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Linear Algebra/Matrix Equation

Diagonal Matrix
A diagonal matrix, $$A$$, is a square matrix in which the entries outside of the main diagonal are zero. The main diagonal of a square matrix consists of the entries which run from the top left corner to the bottom right corner.

In the example below the main diagonal are $$a_{11}, a_{22}, ..., a_{nn}\!$$
 * $$ \quad A=\begin{bmatrix}a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn}\end{bmatrix}$$

Identity Matrix
The identity matrix, with a size of n, is an n-by-n square matrix with ones on the main diagonal and zeros elsewhere. It is commonly denoted as $$I_n$$, or simply by I if the size is immaterial or can be easily determined by the context.
 * $$I_1=[1] \quad I_2=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} \quad I_3=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \quad I_n=\begin{bmatrix}1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1\end{bmatrix}$$

The most important property of the identity matrix is that, when multiplied by another matrix, A, the result will be A
 * $$AI_n=A\,$$ and $$I_n A=A\,$$.

title: Linear Algebra/Matrix Equation


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Linear Algebra/Vector Spaces And Subspaces

A vector space is a way of generalizing the concept of a set of vectors. For example, the complex number 2+3i can be considered a vector, since in some way it is the vector $$\begin{bmatrix} 2 \\ 3\end{bmatrix}$$.

The vector space is a "space" of such abstract objects, which we term "vectors".

Some familiar friends
Currently in our study of vectors we have looked at vectors with real entries: $$\mathbb{R}^2, \mathbb{R}^3, ..., \mathbb{R}^n$$, and so on. These are all vector spaces. The advantage we gain in abstracting to vector spaces is a way of talking about a space without any particular choice of objects (which define our vectors), operations (which act on our vectors), or coordinates (which identify our vectors in the space). Further results may be applied to more general spaces which may have infinite dimension, such as in Functional Analysis.

Notations and concepts
We write a vector, like we have before, bold, but you should write these on paper underlined or with an arrow on top. So we write $$\mathbf{v}=\begin{pmatrix} 2 \\ 3\end{pmatrix}$$ for that vector.

When we multiply a vector by a scalar number, we usually ascribe it a Greek letter, writing λv for the multiplication of v by a scalar λ. We write addition and subtraction of vectors as we have been doing before, x+y for the sum of vectors x and y.

With scalar multiplication and adding vectors, we can move to our definition of a vector space.

When we refer to an operation being 'closed' in a definition, we are saying that the result of the operation does not violate our definition. For example, if we are looking at the set of all integers, we can say that it is closed under addition, because adding any integers results in something inside the set of integers. However the set of integers is not closed under division, because dividing 3 by 2 (for example) doesn't result in a member of the set of integers.

Definition
A vector space is a nonempty set V of objects, called vectors on which are defined two operations, called vector addition and scalar multiplication, respectively, are defined such that, for $$x,y\in V$$ and α$$ \in F$$, x+y and αx are well defined elements of V with the following properties:
 * commutativity of addition: x+y=y+x
 * associativity of addition: x+(y+z)=(x+y)+z
 * additive identity: there is a vector 0 such that 0+x=x for all x
 * additive inverse: for each vector x, there exists another vector y such that x+y =0
 * scalar associativity: α(βx) = (αβ)x
 * scalar distributivity: (α + β)x=αx+βx
 * vector distributivity: α(x+y)=αx+αy
 * scalar identity: 1x=x

Alternative Definition
People who are familiar with group theory and field theory may find the following alternative definition more compact:

 * $$(V, +)$$ is an Abelian group.
 * $$\forall \alpha \in F, (\forall x, y \in V), \alpha (x+y) = \alpha x + \alpha y$$
 * $$\forall \alpha, \beta \in F, \forall x\in V, (\alpha + \beta)x = \alpha x + \beta x$$
 * $$\forall \alpha, \beta \in F, \forall x\in V, (\alpha \beta)x = \alpha (\beta x)$$
 * $$\forall x\in V, 1x = x$$

Subspaces
A subspace is a vector space inside a vector space. When we look at various vector spaces, it is often useful to examine their subspaces.

The subspace S of a vector space V is that S is a subset of V and that it has the following key characteristics
 * S is closed under scalar multiplication: if λ∈R, v∈S, λv∈S
 * S is closed under addition: if u, v ∈ S, u+v∈S.
 * S contains 0, the zero vector.

Any subset with these characteristics is a subspace.

Examples
Let us examine some subspaces of some familiar vector spaces, and see how we can prove that a certain subset of a vector space is in fact a subspace.

The trivial subspace
In R2, the set containing the zero vector ({0}) is a subspace in R2.

Scalar multiplication closure: a 0=0 for all a in R

Addition closure: 0+0=0. Since 0 is the only member of the set so we only need to check 0

Zero vector: 0 is the only member of the set and it is the zero vector.

A slightly less trivial subspace
In R2, the set V of all vectors from R2 of the form (0,α) where α is in R is a subspace

Scalar multiplication closure: a (0,α) = (0,a α) and a α is in R

Addition closure: (0,α) +(0,β) =(0, α + β) and α + β is in R

Zero vector: taking α to be zero in our definition of (0, α) in V we get the zero vector (0,0)

A whole family of subspaces
Pick any number from R, say ρ. Then the set V of all vectors of the form (α, ρα) is a subspace of R2

Scalar multiplication closure: a (α, ρα) = (aα, ρaα) which is in V.

Addition closure: (α, ρα) +(β, ρβ) =(α + β, ρα + ρβ) = (α+β, ρ(α+β)) which is in V

Zero vector: taking α to be zero in our definition we get (0, ρ0) = (0,0) in V.

That means V2 = the set of all vectors of the form (α,2α) is a subspace of R2

and V3 = the set of all vectors of the form (α,3α) is a subspace of R2

and V4 = the set of all vectors of the form (α,4α) is a subspace of R2

and V5 = the set of all vectors of the form (α,5α) is a subspace of R2

and Vπ = the set of all vectors of the form (α,πα) is a subspace of R2

and V√2 = the set of all vectors of the form $$(\alpha,\sqrt{2}\alpha)$$ is a subspace of R2

As you can see, even a simple vector space like R2 can have many different subspaces.

Linear Combinations
Definition: Assume $$V$$ is a vector space over a field $$(F, +, \cdot)$$ and $$S$$ is a nonempty subset of $$V$$. Then a vector $$x\in V$$ is said to be a linear combination of elements of $$S$$ if there exists a finite number of elements $$y_1, y_2, ..., y_n\in S$$ and $$a_1, a_2, ..., a_n\in F$$ such that $$ x = a_1y_1 + a_2y_2 + ... a_ny_n$$.

Spans
Definition: Assume $$V$$ is a vector space over a field $$(F, +, \cdot)$$. The set of all linear combinations of $$y_1, y_2, ..., y_n\in V$$ is called the span of $$y_1, y_2, ..., y_n$$. This is sometimes denoted by $$Span(y_1, y_2, ..., y_n)$$.

Note that $$Span(y_1, y_2, ..., y_n\in V)$$ is a subspace of $$V$$.

Proof: Consider closure under addition and scalar multiplication for two vectors, x and y, in the span of the vectors $${v_1, v_2, ..., v_n}.$$

$$x = a_1v_1 + a_2v_2 + ... + a_nv_n$$

$$y = b_1v_1 + b_2v_2 + ... + b_nv_n$$

$$x+y = (a_1+b_1)*v_1 + (a_2+b_2)*v_2 + ... + (a_n+b_n)*v_n$$, which is also contained in the set.

$$k*x = (k*a_1)v_1 + (k*a_2)v_2 + ... + (k*a_n)v_n$$, which is also contained in the set.

Spanning Sets
Definition: Assume $$V$$ is a vector space over a field $$(F, +, \cdot)$$ and $$y_1, y_2, ..., y_n$$ are vectors in such a vector space. The set $${y_1, y_2, ..., y_n}$$ is a spanning set for the vector space $$V$$ if and only if every vector in $$V$$ is a linear combination of $$y_1, y_2, ..., y_n$$. Alternately, $$\forall x\in V, (\exists a_1, a_2, ..., a_n\in F), x = a_1y_1 + a_2y_2 + ... +a_ny_n$$

Linear Independence
Definition: Assume $$V$$ is a vector space over a field $$(F, +, \cdot)$$ and $$S = \{x_1, x_2, ..., x_n\}$$ is a finite subset of $$V$$. Then we say $$S$$ is linearly independent if $$a_1x_1+a_2x_2+...a_nx_n = 0$$ implies $$a_1 = a_2 = ... = a_n = 0$$. Linear independence is a very important topic in Linear Algebra. The definition implies that linearly dependent vectors may form the nulvector as a non-trivial combination, from which we may conclude that one of the vectors can be expressed as a linear combination of the others.

If we have a vector space V spanned by 3 vectors $${v_1, v_2, v_3}$$ we say that v1, v2, and v3 are linearly dependent if there is a combination of one or two of them that can produce a third. For instance, if one of the following equations:

 * $$a_1v_1 + a_2v_2 = v_3$$
 * $$a_2v_2 + a_3v_3 = v_1$$
 * $$a_1v_1 + a_3v_3 = v_2$$

can be satisfied, then the vectors in V are said to be linearly dependant.

How can we test for linear independence? The definition sets it out to us: If V is a vector space spanned by 3 vectors of length N:

 * $$\tilde{V} = [v_1, v_2, v_3]$$

and we try to test whether these 3 vectors are linearly independent, we form the equations:

 * $$a_1v_1+a_2v_2+a_3v_3=0\,$$

and solve them. If the only solution is

 * $$a_1=a_2=a_3=0,\,$$

then the 3 vectors are linearly independent. If there is another solution they are linearly dependent.

?????? We can say that for V to be linearly independent it must satisfy this condition:

 * $$\tilde{V}\bar{a} = 0$$

Where we are using 0 to denote the null vector in V. If $$\tilde{V}$$ is square and invertable, we can solve this equation directly:

 * $$\tilde{V}^{-1}\tilde{V}\bar{a} = \bar{a} = \tilde{V}^{-1} \cdot 0$$
 * $$\bar{a} = 0$$

And if we know that $$\bar{a}$$ is zero, then we know that the system is linearly independent. If, however, $$\tilde{V}$$ is not square, or if it is not invertable, we can try the following technique:

Multiply through by the transpose matrix:
 * $$\tilde{V}^T\tilde{V}\bar{a} = 0$$

Find the inverse of $$[\tilde{V}^T\tilde{V}$$, and multiply through by the inverse:
 * $$[\tilde{V}^T\tilde{V}]^{-1}\tilde{V}^T\tilde{V}\bar{a} = [\tilde{V}^T\tilde{V}]^{-1} \cdot 0$$

Cancel the terms:
 * $$\bar{a} = \tilde{V}^T\tilde{V} \cdot 0$$

And our conclusion:
 * $$\bar{a} = 0$$

This again means that V is linearly independent.

Span
A span is the set of all possible linear combination of elements of a vector space.(Linear Closure)

Basis
A basis for a vector space is the least amount of linearly independent vectors that can be used to describe the vector space completely. The most common basis vectors are the kronecker vectors, also called canonical basis:

 * $$ \hat i = \begin{pmatrix}1 \\ 0 \\ 0\end{pmatrix}\quad ; \quad \hat j = \begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}\quad ; \quad \hat k = \begin{pmatrix}0 \\ 0 \\ 1\end{pmatrix}$$

In the cartesian graphing space, we say an ordered triple of coordinates is defined as:

 * $$v = \begin{pmatrix}x \\ y \\ z\end{pmatrix}$$

And we can make any point (x, y, z) by combining the kronecker basis vectors:

 * $$\begin{pmatrix}x \\ y \\ z\end{pmatrix} = x\hat i + y\hat j + z\hat k$$

Some theorems:
 * A basis $$\mathcal{B}(v_1, \ldots, v_n)$$ of a vector space V has the maximal number of linearly independent vectors.
 * (Converse) A maximal number of linearly independent vectors in a vector space is a basis.

Bases and Dimension
If a vector space V is such that: it contains a linearly independent set B of N vectors,and any set of N + 1 or more vectors in V is linearly dependent,

then V is said to have dimension N, and B is said to be a  basis of V.

TODO
Tell about what is a basis in a vector space and about coordinate transformations. (this article contains an abstract definition of a basis which is a generalization of a basis in vector space and can be used as the foundation to explain about bases and coordinate transformations.)

Discuss the geometry of subspaces (points, lines, planes, hypersurfaces) and connect them to the geometry of solutions of linear systems. Connect the algebra of subspaces and linear combinations of vectors to the algebra of linear systems.

RECOMMENDATION: extend it to 4D

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Linear Algebra/Introduction to Matrices and Determinants

The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers).

Matrices
Informally an m×n matrix (plural matrices) is a rectangular table of entries from a field (that is to say that each entry is an element of a field). Here m is the number of rows and n the number of the columns in the table. Those unfamiliar with the concept of a field, can for now assume that by a field of characteristic 0 (which we will denote by F) we are referring to a particular subset of the set of complex numbers.

An m×n matrix (read as m by n matrix), is usually written as:

 * $$A=\left(\begin{matrix}

a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn} \end{matrix}\right)$$

The $$i^{th}$$ row is an element of $F^n$, showing the n components $$\begin{pmatrix}a_{i1} & a_{i2} &\cdots a_{in} \end{pmatrix}$$. Similary the $$j^{th}$$ column is an element of $$F^m$$ showing the m components $$\begin{pmatrix}a_{1j}\\a_{2j}\\\vdots\\a_{mj}\end{pmatrix}$$.

Here m and n are called the dimensions of the matrix. The dimensions of a matrix are always given with the number of rows first, then the number of columns. It is also said that an m by n matrix has an order of m×n.

Formally, an m×n matrix M is a function $$M:A\rightarrow F$$ where A = {1,2...m} × {1,2...n} and F is the field under consideration. It is almost always better to visualize a matrix as a rectangular table (or array) then as a function.

A matrix having only one row is called a row matrix (or a row vector) and a matrix having only one column is called a column matrix (or a column vector). Two matrices of the same order whose corresponding entries are equal are considered equal. The (i,j)-entry of the matrix (often written as $$A_{ij}$$ or $$A_{i,j}$$) is the element at the intersection of the $$i^{th}$$ row (from the top) and the $$j^{th}$$ column (from the left).

For example,

 * $$\begin{pmatrix} 3 & 4 & 8 \\

2 & 7 & 11 \\                       1 & 1 & 1  \end{pmatrix}$$

is a 3×3 matrix (said 3 by 3). The 2nd row is $$\begin{pmatrix}2 & 7 & 11 \end{pmatrix}$$ and the 3rd column is $$\begin{pmatrix} 8\\ 11 \\ 1\end{pmatrix}$$. The (2,3) entry is the entry at intersection of the 2nd row and the 3rd column, that is 11.

Some special kinds of matrices are:

 * A square matrix is a matrix which has the same number of rows and columns. A diagonal matrix is a matrix with non zero entries only on the main diagonal (ie at $$A_{i,i}$$ positions).

\begin{cases} 1, & i=j \\ 0, & i \neq j \end{cases}$$
 * The unit matrix or identity matrix In, is the matrix with elements on the diagonal set to 1 and all other elements set to 0. Mathematically, we may say that for the identity matrix $$I_{i,j}$$ (which is usually written as $$\delta_{i,j}$$ and called Kronecker's delta) is given by: $$ \delta_{i,j} =

For example, if n = 3:

I_3 = \begin{bmatrix} 1 & 0 & 0 \\   0 & 1 & 0 \\    0 & 0 & 1  \end{bmatrix} .$$

\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}^{\mathrm{T}} \!\! \;\! = \, \begin{bmatrix} 1 & 3 & 5\\ 2 & 4 & 6 \end{bmatrix} \; $$
 * The transpose of an m-by-n matrix A is the n-by-m matrix AT formed by turning rows into columns and columns into rows, i.e. $$A_{i,j} = A^T_{j,i} \forall i,j$$. An example is $$

1 & 2 & 3\\ 2 & 4 & -5\\ 3 & -5 & 6\end{bmatrix}$$
 * A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if $$A^{\mathrm{T}} = A.\,$$. An example is $$\begin{bmatrix}

0 & -3 & 4\\ 3 & 0 & -5\\ -4 & 5 & 0\end{bmatrix}$$
 * A square matrix whose transpose is equal to its negative is called skew-symmetric matrix; that is, A is skew-symmetric if $$A^{\mathrm{T}} = -A.\,$$. An example is $$\begin{bmatrix}

Properties of these matrices are developed in the exercises.

Determinants
To define a determinant of order n, suppose there are n2 elements of a field sij where i and j are less than or equal to n. Define the following function (this function is important in the definition):

S(a1,a2,a3,...,an)=# of reversals, meaning the number of times an1n2, for each possible combination.

Suppose you have a permutation of numbers from 1 to n {a1,a2,a3,...,an). Then define a term of the determinant to be equal to (-1)S(a1,a2,a3,...,an)s1a1,s2a2,s3a3,...,snan. The sum of all possible terms (i. e. through all possible permutations) is called the determinant.

Theorem
Definition: The transpose of a matrix A, AT is the matrix resulting when the columns and rows are interchanged i. e. the matrix sji when A is the matrix sij A matrix and its transpose have the same determinant:
 * $$\det(A^\top) = \det(A). \,$$

Proof
All terms are the same, and the signs of the terms are also unchanged since all reversals remain reversals. Thus, the sum is the same.

Theorem
Interchanging two rows (or columns) changes the sign of the determinant:
 * $$\det \begin{bmatrix} \cdots \\ \mbox{row A} \\ \cdots \\ \mbox{row B} \\ \cdots \end{bmatrix} = - \det \begin{bmatrix}\cdots \\ \mbox{row B} \\ \cdots \\ \mbox{row A} \\ \cdots \end{bmatrix} $$.

Proof
To show this, suppose two adjacent rows (or columns) are interchanged. Then any reversals in a term would not be affected except for the reversal of the elements of that term within that row (or column), in which case adds or subtracts a reversal, thus changing the signs of all terms, and thus the sign of the matrix. Now, if two rows, the ath row and the (a+n)th are interchanged, then interchange successively the ath row and (a+1)th row, and then the (a+1)th row and (a+2)th row, and continue in this fashion until one reaches the (a+n-1)th row. Then go backwards until one goes back to the ath row. This has the same effect as switching the ath row and the (a+n)th rows, and takes n-1 switches for going forwards, and n-2 switches for going backwards, and their sum must then be an odd number, so it multiplies by -1 an odd number of times, so that its total effect is to multiply by -1.

Corollary
A determinant with two rows (or columns) that are the same has the value 0. Proof: This determinant would be the additive inverse of itself since interchanging the rows (or columns) does not change the determinant, but still changes the sign of the determinant. The only number for which it is possible is when it is equal to 0.

Theorem
It is linear on the rows and columns of the matrix.
 * $$\det \begin{bmatrix} \ddots & \vdots & \ldots \\ \lambda a_1 + \mu b_1 & \cdots & \lambda a_n + \mu b_n \\   \cdots & \vdots & \ddots \end{bmatrix} = \lambda \det \begin{bmatrix}  \ddots & \vdots & \cdots \\ a_1 & \cdots & a_n \\   \cdots & \vdots & \ddots \end{bmatrix} + \mu \det \begin{bmatrix}  \ddots & \vdots & \cdots \\  b_1 & \cdots & b_n \\   \cdots & \vdots & \ddots \end{bmatrix}$$

Proof
The terms are of the form a1...$$\lambda a + \mu b$$...an. Using the distributive law of fields, this comes out to be a1...$$\lambda a$$...an + a1...$$\mu b$$...an, an thus its sum of such terms is the sum of the two determinants:

$$\lambda \det \begin{bmatrix} \ddots & \vdots & \cdots \\ a_1 & \cdots & a_n \\   \cdots & \vdots & \ddots \end{bmatrix} + \mu \det \begin{bmatrix}  \ddots & \vdots & \cdots \\  b_1 & \cdots & b_n \\   \cdots & \vdots & \ddots \end{bmatrix}$$

Corollary
Adding a row (or column) times a number to another row (or column) does not affect the value of a determinant.

Proof
Suppose you have a determinant A with the kth column added by another column times a number: $$ \begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{1k}+\mu a_{1b} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2k}+\mu a_{2b} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \ldots & a_{3k}+\mu a_{3b} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\  a_{n1} & a_{n2} & a_{n3} & \ldots & a_{nk}+\mu a_{nb} & \ldots & a_{nn} \end{bmatrix}$$

where akb are elements of another column. By the linear property, this is equal to

$$ \begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{1k} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2k} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \ldots & a_{3k} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\  a_{n1} & a_{n2} & a_{n3} & \ldots & a_{nk} & \ldots & a_{nn} \end{bmatrix} +  \begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & \mu a_{1b} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & \mu a_{2b} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \ldots & \mu a_{3b} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\  a_{n1} & a_{n2} & a_{n3} & \ldots & \mu a_{nb} & \ldots & a_{nn} \end{bmatrix}$$

The second number is equal to 0 because it has two columns that are the same. Thus, it is equal to $$ \begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{1k} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2k} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \ldots & a_{3k} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\  a_{n1} & a_{n2} & a_{n3} & \ldots & a_{nk} & \ldots & a_{nn}\end{bmatrix}$$

which is the same as the matrix A.

 * It is easy to see that $$\det(rI_n) = r^n \,$$ and thus
 * $$\det(rA) = \det(rI_n \cdot A) = r^n \det(A) \,$$ for all $$n$$-by-$$n$$ matrices $$A$$ and all scalars $$r$$.

 * A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In particular, if A is a matrix over a field such as the real or complex numbers, then A is invertible if and only if det(A) is not zero. In this case we have

 * $$\det(A^{-1}) = \det(A)^{-1}. \,$$

Expressed differently: the vectors v1,...,vn in Rn form a basis if and only if det(v1,...,vn) is non-zero.

The determinants of a complex matrix and of its conjugate transpose are conjugate:
 * $$\det(A^*) = \det(A)^*. \,$$


text:
Linear Algebra/Inner Product Length and Orthogonality

Cauchy-Schwarz inequality
The Cauchy-Schwarz inequality states that the magnitude of the inner product of two vectors is less than or equal to the product of the vector norms, or: $$| \langle x,y \rangle | \le  \| x \| \| y \| $$.

Definition
For any vectors $$x$$ and $$y$$ in an inner product space $$V$$, we say $$x$$ is orthogonal to $$y$$, and denote it by $$x \bot y$$, if $$\langle x,y \rangle =0$$.

How to orthogonalize a basis
Suppose to be on a vector space V with a scalar product (not necessarily positive-definite), Problem: Construct an orthonormal basis of V starting by a random basis { v1, ... }. Solution: Gram-Schmidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.


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Linear Algebra/The Inverse of a Matrix

An n-by-n matrix A is the inverse of n-by-n matrix B (and B the inverse of A) if BA = AB = I, where I is an identity matrix.

The inverse of an n-by-n matrix can be calculated by creating an n-by-2n matrix which has the original matrix on the left and the identity matrix on the right. Row reduce this matrix and the right half will be the inverse. If the matrix does not row reduce completely (i.e., a row is formed with all zeroes as its entries), it does not have an inverse.

Example
Let $$ \mathrm{A} = \begin{bmatrix} 1 & 4 & 4\\ 2 & 5 & 8\\   3 & 6 & 9 \end{bmatrix}$$

We begin by expanding and partitioning A to include the identity matrix, and then proceed to row reduce A until we reach the identity matrix on the left-hand side.

\begin{bmatrix} 1 & 4 & 4 & \big| & 1 & 0 &0\\ 2 & 5 & 8 & \big| & 0 & 1 &0\\ 3 & 6 & 9 & \big| & 0 & 0 &1 \end{bmatrix}

\rightsquigarrow

\begin{bmatrix} 1 & 4 & 4 & \big| & 1 & 0 &0\\ 0 & -3 & 0 & \big| & -2 & 1 &0\\ 0 & -6 & -3 & \big| & -3 & 0 &1 \end{bmatrix}

\rightsquigarrow

\begin{bmatrix} 1 & 4 & 4 & \big| & 1 & 0 &0\\ 0 & 1 & 0 & \big| & 2/3 & -1/3 &0\\ 0 & -6 & -3 & \big| & -3 & 0 &1 \end{bmatrix}

\rightsquigarrow

$$

\begin{bmatrix} 1 & 0 & 4 & \big| & -5/3 & 4/3 &0\\ 0 & 1 & 0 & \big| & 2/3 & -1/3 &0\\ 0 & 0 & -3 & \big| & 1 & -2 &1 \end{bmatrix}

\rightsquigarrow

\begin{bmatrix} 1 & 0 & 4 & \big| & -5/3 & 4/3 &0\\ 0 & 1 & 0 & \big| & 2/3 & -1/3 &0\\ 0 & 0 & 1 & \big| & -1/3 & 2/3 &-1/3 \end{bmatrix}

\rightsquigarrow

\begin{bmatrix} 1 & 0 & 0 & \big| & -1/3 & -4/3 &4/3\\ 0 & 1 & 0 & \big| & 2/3 & -1/3 &0\\ 0 & 0 & 1 & \big| & -1/3 & 2/3 &-1/3 \end{bmatrix}

$$ The matrix $$ \mathrm{B} = \begin{bmatrix} -1/3 & -4/3 & 4/3\\ 2/3 & -1/3 & 0\\   -1/3 & 2/3 & -1/3 \end{bmatrix} $$ is then the inverse of the original matrix A.

title: Linear Algebra/The Inverse of a Matrix


text:
Linear Algebra/Inner Product Spaces

Recall that in your study of vectors, we looked at an operation known as the dot product, and that if we have two vectors in Rn, we simply multiply the components together and sum them up. With the dot product, it becomes possible to introduce important new ideas like length and angle. The length of a vector,$$\mathbf{a}$$, is just $$||\mathbf{a}||=\sqrt{\mathbf{a}\cdot\mathbf{a}}$$. The angle between two vectors,$$\mathbf{a}$$ and $$\mathbf{b}$$, is related to the dot product by

$$\cos{\theta} = \frac{\mathbf{a}\cdot\mathbf{b}} {||\mathbf{a}||||\mathbf{b}||}$$

It turns out that only a few properties of the dot product are necessary to define similar ideas in vector spaces other than Rn, such as the spaces of $$m\times n$$ matrices, or polynomials. The more general operation that will take the place of the dot product in these other spaces is called the "inner product".

The inner product
Say we have two vectors:
 * $$\mathbf{a}=\begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix}, \mathbf{b}=\begin{pmatrix} 6 \\ 3 \\ 0 \end{pmatrix}$$

If we want to take their dot product, we would work as follows
 * $$\mathbf{a}\cdot\mathbf{b} = a_1b_1+a_2b_2+a_3b_3 = (2)(6)+(1)(3)+(4)(0) = 15$$

Because in this case multiplication is commutative, we then have a·b = b · a.

But then, we observe that
 * $$\mathbf{v}\cdot(\alpha\mathbf{a}+\beta\mathbf{b}) = \alpha\mathbf{v}\cdot\mathbf{a} + \beta\mathbf{v}\cdot\mathbf{b}$$

much like the regular algebraic equality v(aA+bB)=avA+bvB. For regular dot products this is true since, for R3, for example, one can expand both sides out to obtain

\begin{matrix} (\alpha v_1a_1+\beta v_1b_1)+(\alpha v_2a_2+\beta v_2b_2)+(\alpha v_2a_2+\beta v_2b_2)=\\ (\alpha v_1a_1+\alpha v_2a_2+\alpha v_3a_3)+(\beta v_1b_1+\beta v_2b_2+\beta v_3+b_3) \end{matrix}$$

Finally, we can notice that v·v is always positive or greater than zero - checking this for R3 gives this as
 * $$\mathbf{v}\cdot\mathbf{v}=v_1^2+v_2^2+v_3^2$$

which can never be less than zero since a real number squared is positive. Note that v·v = 0 if and only if v = 0.

In generalizing this sort of behaviour, we want to keep these three behaviours. We can then move on to a definition of a generalization of the dot product, which we call the inner product. An inner product of two vectors in some vector space V, written < x, y > is a function that maps V×V to R, which obeys the property that
 * < x, y > = < y, x >
 * < v, αa+βb > = α < v, a > + β < v, b >
 * < a, a > ≥ 0, < a, a > = 0 iff a = 0.

The vector space V and some inner product together are known as an inner product space.

title: Linear Algebra/Inner Product Spaces


text:
Linear Algebra/Row Reduction and Echelon Forms

Introduction
Many of the problems you will solve in linear algebra require that a matrix be converted into one of two forms, the Row Echelon Form (ref) and its stricter variant the Reduced Row Echelon Form  (rref). These two forms will help you see the structure of what a matrix represents. If the matrix represents a system of linear equations, these forms allow one to write the solutions to the system. Every Computer Algebra System and most scientific or graphing calculators have commands which produce these forms for any matrix. The commands are often of the form rref(A), for example.

The Row Echelon Form
A rectangular matrix is in row echelon form if it has the following three properties: This matrix is in row echelon form:
 * All nonzero rows are above any rows of all zeros
 * Each leading entry of a row is in a column to the right of the leading entry of the row above it
 * All entries of a column below a leading entry are zeros
 * $$\begin{bmatrix} 1 && 0 && 3 && 3\\ 0 && 1 && 0 && 4 \\ 0 && 0 && 0 && 1 \end{bmatrix}$$

The Reduced Row Echelon Form
If a matrix in echelon form satisfies the following conditions, then it is in reduced row echelon form: This matrix is in reduced row echelon form:
 * The matrix is in row-echelon form
 * Each leading 1 is the only nonzero entry in its column
 * $$\begin{bmatrix} 1 && 0 && 0\\ 0 && 1 && 0 \\ 0 && 0 &&1 \end{bmatrix}$$

and the rref of the matrix in the previous section is:
 * $$\begin{bmatrix} 1 && 0 && 3&&0\\ 0 && 1 && 0 &&0\\ 0 && 0 && 0 &&1

\end{bmatrix}$$

Changing A Matrix Into REF Or RREF Form
A system of linear equations can be solved by reducing its augmented matrix into reduced echelon form.

A matrix can be changed to its reduced row echelon form, or row reduced to its reduced row echelon form using the elementary row operations. These are:
 * 1) Interchange one row of the matrix with another of the matrix.
 * 2) Multiply one row of the matrix by a nonzero scalar constant.
 * 3) Replace the one row with the one row plus a constant times another row of the matrix.

For example, given the following linear system with corresponding augmented matrix:  $$      3x_2 -  6x_3 + 6x_4 + 4x_5 = -5$$

$$3x_1 - 7x_2 + 8x_3 - 5x_4 + 8x_5 =  9$$

$$3x_1 - 9x_2 + 12x_3 - 9x_4 + 6x_5 = 15$$

 $$\begin{bmatrix} 0 & 3 & -6 & 6 & 4 & -5 \\ 3 & -7 & 8 & -5 & 8 & 9 \\ 3 & -9 & 12 & -9 & 6 & 15 \end{bmatrix}$$ To solve this system, the matrix has to be reduced into reduced echelon form.

Step 1: Switch row 1 and row 3. All leading zeros are now below non-zero leading entries. $$\begin{bmatrix} 3 & -9 & 12 & -9 & 6 & 15 \\ 3 & -7 & 8 & -5 & 8 & 9 \\ 0 & 3 & -6 & 6 & 4 & -5 \end{bmatrix}$$

Step 2: Set row 2 to row 2 plus (-1) times row 1. In other words, subtract row 1 from row 2. This will eliminate the first entry of row 2.

$$\begin{bmatrix} 3 & -9 & 12 & -9 & 6 & 15 \\ 0 & 2 & -4 & 4 & 2 & -6 \\ 0 & 3 & -6 & 6 & 4 & -5 \end{bmatrix}$$

Step 3: Multiply row 2 by 3 and row 3 by 2. This will eliminate the first entry of row 3.

$$\begin{bmatrix} 3 & -9 & 12 & -9 & 6 & 15 \\ 0 & 6 & -12 & 12 & 6 & -18 \\ 0 & 6 & -12 & 12 & 8 & -10 \end{bmatrix}$$

Step 4: Set row 3 to row 3 plus (-1) times row 2. In other words, subtract row 2 from row 3. This will eliminate the second entry of row 3.

$$\begin{bmatrix} 3 & -9 & 12 & -9 & 6 & 15 \\ 0 & 6 & -12 & 12 & 6 & -18 \\ 0 & 0 & 0 & 0 & 2 & 8 \end{bmatrix}$$

Step 5: Multiply each row by the reciprocal of its first non-zero value. This will make every row start with a 1. $$\begin{bmatrix} 1 & -3 & 4 & -3 & 2 & 5 \\ 0 & 1 & -2 & 2 & 1 & -3 \\ 0 & 0 & 0 & 0 & 1 & 4 \end{bmatrix}$$

The matrix is now in row echelon form: All nonzero rows are above any rows of all zeros (there are no zero rows), each leading entry of a row is in a column to the right of the leading entry of the row above it and all entries in a column below a leading entry are zeros.

As can and will be shown later, from this form one can observe that the system has infinitely many solutions. To get those solutions, the matrix is further reduced into reduced echelon form.

Step 6: Set row 2 to row 2 plus (-1) times row 3 and row 1 to row 1 plus (-2) times row 3. This will eliminate the entries above the leading entry of row 3.

$$\begin{bmatrix} 1 & -3 & 4 & -3 & 0 & -3 \\ 0 & 1 & -2 & 2 & 0 & -7 \\ 0 & 0 & 0 & 0 & 1 & 4 \end{bmatrix}$$

Step 7: Set row 1 to row 1 plus 3 times row 2. This eliminates the entry above the leading entry of row 2.

$$\begin{bmatrix} 1 & 0 & -2 & 3 & 0 & -24 \\ 0 & 1 & -2 & 2 & 0 & -7 \\ 0 & 0 & 0 & 0 & 1 & 4 \end{bmatrix}$$

This is a reduced echelon form, since the leading entry in each nonzero row is 1 and each leading 1 is the only nonzero entry in its column.

From this the solution of the system can be read:

$$x_1 - 2x_3 + 3x_4 = -24$$

$$x_2 - 2x_3 + 2x_4 = -7$$

$$x_5 = 4$$

Those equations can be solved for $$x_1$$, $$x_2$$ and $$x_5$$:

$$x_1 = 2x_3 - 3x_4 -24$$

$$x_2 = 2x_3 - 2x_4 -7$$

$$x_5 = 4$$

This is the solution of the system. The variables $$x_3$$ and $$x_4$$ can take any value and are so called free variables. The solution is valid for any $$x_3$$ and $$x_4$$.

Understanding The Two Forms
Any nonzero matrix may be row reduced into more than one matrix in echelon form, by using different sequences of row operations. However, no matter how one gets to it, the reduced row echelon form of every matrix is unique.

If matrix A is row equivalent to an echelon matrix B, we call matrix B an echelon form of A, if B is in reduced echelon form, we call B the reduced echelon form of A.

Pivot Positions
A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position.


text:
Perl Programming/Basic variables

In Perl, there are five different types of variables, each indicated by its own sigil character: scalars ($), arrays (@), hashes (%), subroutines (&), and typeglobals (*). We first introduce scalars, as they are easiest to understand.

Scalar variables
Although they have a complicated name, a scalar variable is just a way of storing a value for later use. For example, consider:

If we wanted to change the name of the user, we can just change the string. However, if this program were hundreds of lines long and the name was used several times, trying to find the strings throughout the program would be more fuss than it was worth. Instead, it is common to use variables for any values that might change.

This new program gives exactly the same output as the previous one. There are several new things about this program, the most obvious being the line

 * my $firstname = "John";
 * my Tells the perl interpreter to give you a new variable. Technically, this is optional but highly recommended.
 * $firstname Tells the perl interpreter that you want a scalar variable called firstname
 * = Tells the perl interpreter to assign the value on it's right hand side to the variable on the left hand side
 * "John" Tells the perl interpreter that every time you use $firstname you want it to use the string John.

While using variables to contain information that you have programmed is all very well, variables are used to store data that can change while your program runs, as their name implies.

You should notice that when we change the value of $firstname we do not need to use the my operator. This is because we have already told perl to give us the variable, and it is now ours to do what we wish with.

Numbers
Now, while strings are very useful, there is a place in computer programs for numbers as well. A scalar variable can contain either a number or a string.

Perl does not provide any easy way of telling the type of information stored in a scalar variable. Indeed, if you give Perl a string and ask it to do number operation, Perl will automatically convert the string into the number; and vice versa. Here is an example:

If you run this program, you can see that both the first two examples result in 11 being stored to $y. Perl does not convert English words to numbers, though, and if you give a string like <tt>"ten"</tt> it will not turn it to 10, but instead just think it is 0. The last example shows that it actually tries to understand the string as a number, and stops whenever it finds something it doesn't understand.

Raw printing
It is of course possible to output variables without including them in a string.

Exercises
Perl/Variáveis escalares

title: Perl Programming/Basic variables


text:
User:Milhaus

Welcome,

I am Milhaus, come from Czech Republic. I currently study geography in Brno, Czech Republic. You can find more information on my czech Wikipedia profile.

title: User:Milhaus


text:
Econometric Theory/Serial Correlation

There are times, especially in time-series data, that the CLR assumption of $$ corr(\epsilon_t, \epsilon_{t-1})=0 $$ is broken. This is known in econometrics as Serial Correlation or Autocorrelation. This means that $$ corr(\epsilon_t, \epsilon_{t-1}) \ne 0 $$ and there is a pattern across the error terms. The error terms are then not independently distributed across the observations and are not strictly random.

Examples of Autocorrelation
$$ corr(\epsilon_t, \epsilon_{t-1})>0 $$

$$ corr(\epsilon_t, \epsilon_{t-1})<0 $$

Functional Form
When the error term is related to the previous error term, it can be written in an algebraic equation. $$ \epsilon_t = \rho \epsilon_{t-1} + u_t $$ where ρ is the autocorrelation coefficient between the two disturbance terms, and u is the disturbance term for the autocorrelation. This is known as an Autoregressive Process. $$-1< \rho = corr(\epsilon_t, \epsilon_{t-1}) < 1$$ The u is needed within the equation because although the error term is less random, it still has a slight random effect.

Autoregressive model

 * First order Autoregressive Process, AR(1):$$ \epsilon_t = \rho \epsilon_{t-1} + u_t $$
 * This is known as the first order autoregression, due to the error term only depending on the previous error term.
 * nth order Autoregressive Process, AR(n):$$ \epsilon_t = \rho_1 \epsilon_{t-1} + \rho_2 \epsilon_{t-2} + \cdots + \rho_n \epsilon_{t-n} + u_t $$

Moving-average model
The notation MA(q) refers to the moving average model of order q:

 * $$ X_t = \mu + \varepsilon_t + \sum_{i=1}^q \theta_i \varepsilon_{t-i}\,$$

where the θ1, ..., θq are the parameters of the model, μ is the expectation of $$X_t$$ (often assumed to equal 0), and the $$\varepsilon_t$$, $$\varepsilon_{t-1}$$,... are again, white noise error terms. The moving-average model is essentially a finite impulse response filter with some additional interpretation placed on it.

Autoregressive–moving-average model
The notation ARMA(p, q) refers to the model with p autoregressive terms and q moving-average terms. This model contains the AR(p) and MA(q) models,

 * $$ X_t = c + \varepsilon_t + \sum_{i=1}^p \varphi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,$$

Causes of Autocorrelation
$$ corr(\epsilon_t, \epsilon_{t-1}) \ne 0 $$ Spatial Autocorrelation occurs when the two errors are specially and/or geographically related. In simpler terms, they are "next to each." Examples: The city of St. Paul has a spike of crime and so they hire additional police. The following year, they found that the crime rate decreased significantly. Amazingly, the city of Minneapolis, which had not adjusted its police force, finds that they have an increase in the crime rate over the same period.
 * 1) Spatial Autocorrelation
 * Note: this type of Autocorrelation occurs over cross-sectional samples.
 * 1) Inertia/Time to Adjust
 * 2) This often occurs in Macro, time series data. The US interest rate unexpectedly increases and so there is an associated change in exchange rates with other countries. Reaching a new equilibrium could take some time.
 * 3) Prolonged Influences
 * 4) This is again a Macro, time series issue dealing with economic shocks. It is now expected that the US interest rate will increase. ##The associated exchange rates will slowly adjust up-until the announcement by the Federal Reserve and may overshoot the equilibrium.
 * 5) Data Smoothing/Manipulation
 * 6) Using functions to smooth data will bring autocorrelation into the disturbance terms
 * 7) Misspecification
 * 8) A regression will often show signs of autocorrelation when there are omitted variables. Because the missing independent variable now exists in the disturbance term, we get a disturbance term that looks like: $$ \epsilon_t = \beta_2 X_2 + u_t $$ when the correct specification is $$ Y_t = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + u_t $$

Consequences of Autocorrelation
The main problem with autocorrelation is that it may make a model look better than it actually is.

List of consequences

 * 1) Coefficients are still unbiased $$E(\epsilon_t) = 0, cov(X_t, u_t) = 0 $$
 * 2) True variance of $$ \hat{\beta} $$ is increased, by the presence of autocorrelations.
 * 3) Estimated variance of $$ \hat{\beta} $$ is smaller due to autocorrelation (biased downward).
 * 4) A decrease in $$ se(\hat{\beta}) $$ and an increase of the t-statistics; this results in the estimator looking more accurate than it actually is.
 * 5) R² becomes inflated.

All of these problems result in hypothesis tests becoming invalid.

Testing for Autocorrelation

 * 1) While not conclusive, an impression can be gained by viewing a graph of the dependent variable against the error term (namely, a residual scatter-plot).
 * 2) Durbin-Watson test:
 * 3) Assume $$ \epsilon_t = \epsilon_{t-1} \rho + u_t $$
 * 4) Test H(0): ρ = 0 (no AC) against H(1): ρ > 0 (one-tailed test)
 * 5) Test statistic $$ DW = \frac{\sum (\epsilon_t - \epsilon_{t-1})^2} {\sum \epsilon^2} = 2 - 2 \rho $$
 * Any value under D(L) (in the D-W table) rejects the null hypothesis and AC exists.
 * Any value between D(L) and D(W) leaves us with no conclusion of AC.
 * Any value larger than D(W) accepts the null hypothesis and AC does not exist.
 * Note, this is one tail test. To get the other tail. Use 4 - DW as the test stat instead.

title: Econometric Theory/Serial Correlation


text:
Microsoft Certified Professional Developer/Exam 70-529

Microsoft .NET Framework 2.0 Distributed Application Development MSDN Exam Information

Creating and Accessing XML Web Services

 * Create and configure an XML Web service method.
 * Configure a Web service method by using the properties of the WebMethodAttribute attribute.
 * Use discovery files to publish a list of Web services that are installed on a Web server.
 * Build and deploy a Web service application.


text:
Talk:Biblical Studies/Christianity/Living as a Christian

By whom was the neutrality disputed? What was the dispute?

No Dispute Now
Only 3 contributors, now 2 to 1 that the page approaches NPOV, while scripturally based. Thank you. - Athrash | T a l k 01:36, 15 April 2007 (UTC)


text:
Adventist Youth Honors Answer Book/Nature/Seeds

1. What is the main purpose of a seed?
The main purpose of a seed is to grow a new plant, thus propagating the species.

2. What foods were first given to man in the Garden of Eden?
Then God said, "I give you every seed-bearing plant on the face of the whole earth and every tree that has fruit with seed in it. They will be yours for food. Genesis 1:29

3. Identify from a seed or drawing and know the purpose of each of these parts of a seed: seed coat, cotyledon, embryo.

.
 * Seed Coat - The seed coat is the covering which encloses the seed (small embryonic plant, usually with some stored food).
 * Cotyledon - This is a significant part of the embryo within the seed of a plant. Upon germination, the cotyledon becomes the embryonic first leaves of a seedling.
 * Embryo - A seed embryo includes the cotyledon which becomes the plant's first leaves, the epicotyl (not shown), which becomes the shoot above the leaves, the hypocotyl which becomes the plant's stem, and the radicle which becomes the plant's

Gravity
The effect of gravity on the dispersal of seeds and spores is straightforward. Heavier seeds will tend to drop downward from the parent plant, and not by themselves travel very far. Spores, being much lighter, are more influenced by physical movements in the environment, especially those of wind and water, and therefore less strictly subject to the simple motion of gravity (see examples below). Gravity may be sufficient agent for plants growing on steep slopes, but upslope movement of a population can be a problem. The naked seeds of gymnosperms are largely dependent upon gravity for dispersal. Most conifers are long-lived large shrubs or tall trees, thus taking full advantage of gravitational dispersal and allowing for gradual upslope movement of a population. Dispersal of seeds "strictly" by gravity should not overlook storm effects: seeds from a deteriorating cone growing high on a tall, narrow tree will get spread widely during a wind storm (see "Wind" below).

Encasing seeds in a rounded fruit promotes gravity driven movement away from the parent.

 * Ferns
 * Pine trees
 * Spruce trees
 * Fir trees
 * Junipers
 * Cedars
 * Hemlocks

Mechanical dispersal
Numerous species have mechanical means to overcome the tendency of a seed to drop close to its parent. Seedpods are often shaped so that the seeds are flung away from the parent plant with considerable force as the seedpod matures

Examples of fruit with mechanical dispersal mechanisms:
 * Yellow wood sorrel and Touch-me-not – as the seed dries, becomes sensitive to disturbance, ejecting tiny seeds in an explosive discharge. Touch-me-not is named for this behavior.
 * Maple trees - the seeds of the Maple tree are those little "helicopters" that children love to play with. As the seeds falls, the wings cause it to rotate, slowing its descent, and thus allowing a breeze to carry it farther from the parent.  When the seed strikes the ground, it bores into the soil.

Wind
For non-aquatic, terrestrial plants, the wind is an obvious supplier of energy for movement, and many plants clearly take advantage of this fact. This type of seed dispersal is not efficient, but very effective. Perhaps most familiar are the feather-light fiber parachutes with attached achenes that are produced by a number of species of flowers, a well-known example being the dandelion (see right).

 * Milkweed
 * Thistles
 * Dandelions

Water
Plants that grow in water (aquatic and obligate wetland species) are likely to utilize water to disperse their seeds. For example, all mangroves disperse their offspring by water. In one mechanism, the seedling separates from the fruit, leaving its cotyledons behind, and—floating horizontally on the water surface—is carried away by tidal or river flow. After a month or two, the propagated seed turns vertical in the water. Once it "feels" bottom or strands, roots start to develop and leaves appear at the upper end.

A mechanism commonly seen in coastal plants are those that promote flotation of the fruit, allowing the seed to be carried away on the tide or ocean currents. Examples would be:

 * The coconut produces a large, dry, fiber-filled fruit capable of a long survival adrift at sea.
 * Alexandrian laurel or kamani produces a globose fruit that is almost cork-like.

Animals
A significant aspect of plant-animal cooperation involves plants designed to take advantage of animal abilities to move. Some fruit have prickly burrs or spikes that attach themselves to a passing animal's fur or feathers so that the animal will carry them away. Some seeds are contained within a soft fruit that "invites" animals to consume it. These seeds have a tough protective outer-coating so that while the fruit is digested, the seeds will pass through their host's digestive tract intact, and grow wherever they fall. Some seeds are appealing to rodents (such as squirrels) who hoard them in hidden caches, often beneath the surface of the soil, in order to avoid starving during the winter and early spring. Those seeds that are left uneaten have the chance to germinate and grow into a new plant.

Some animals that disperse may also eat the seed.

 * Beggarlice
 * Cockle burr
 * Apple
 * Banana
 * Strawberry
 * Oak (acorns)

5. List from memory ten kinds of seeds that we use for food.
This list does not include the dozens of seeds used as spices (coriander, cumin, etc.), nor does it include seeds which are incidentally eaten as part of a fruit (strawberry, banana). However, you should accept such answers if they are given.
 * 1) Almond
 * 2) Amaranth
 * 3) Barley
 * 4) Brazil nut
 * 5) Buckwheat
 * 6) Cashew
 * 7) Chestnut
 * 8) Chickpea (garbanzo)
 * 9) Cocoa
 * 10) Coconut
 * 11) Corn
 * 12) Cowpea (Blackeye pea)
 * 13) Filbert (Hazelnut)
 * 14) Flax
 * 15) Green bean
 * 16) Lentil
 * 17) Lima bean
 * 18) Macadamia
 * 19) Millet
 * 20) Mustard
 * 21) Navy bean
 * 22) Oat
 * 23) Pea
 * 24) Peanut
 * 25) Pecan
 * 26) Pine nut
 * 27) Pinto bean
 * 28) Pistachio
 * 29) Pomegranate
 * 30) Pumpkin
 * 31) Quinoa
 * 32) Rice
 * 33) Rye
 * 34) Sesame
 * 35) Soybean
 * 36) Sorghum / Milo
 * 37) Sunflower
 * 38) Teff (found in Ethiopia)
 * 39) Walnut
 * 40) Wheat
 * 41) Wild Rice
 * 42) White bean

6. List from memory five kinds of seeds that are used as sources of oil.

 * 1) Coconut
 * 2) Corn
 * 3) Cottonseed
 * 4) Canola oil (a variety of rapeseed oil)
 * 5) Grapeseed
 * 6) Olive
 * 7) Palm
 * 8) Peanut
 * 9) Safflower
 * 10) Sesame
 * 11) Soybean
 * 12) Sunflower

7. List from memory five kinds of seeds that are used for spices.

 * 1) Anise
 * 2) Caraway
 * 3) Cardamom
 * 4) Cocoa
 * 5) Coriander
 * 6) Cumin
 * 7) Dill
 * 8) Fennel
 * 9) Nutmeg
 * 10) Mustard
 * 11) Vanilla
 * 12) pimento

Requirements for seed germination
Seed germination depends on many factors, both internal and external. The most important external factors include: water, oxygen, temperature, and the correct soil conditions. Every variety of seed requires a different set of variables for successful germination. This depends greatly on the individual seed variety and is closely linked to the ecological conditions in the plants' natural habitat.

Water
Germination requires moist conditions. Mature seeds are usually very dry and need to take up significant amounts of water before they can "come back to life." The uptake of water into seeds leads to a marked swelling. The pressure caused by water aids in cracking the seed coat for germination. When seeds are formed, most plants store large amounts of food, such as starch, proteins, or oils, for the embryo inside the seed. When the seed absorbs water,it stores food resources and allows the seedling to germinate and grow until it reaches the light. Once the seedling starts growing, it requires a continuous supply of water and nutrients.

Oxygen
Most seeds respond best when water levels are enough to moisten the seeds but not soak them, and when oxygen is readily available. Once the seed coat is cracked, the germinating seedling requires oxygen. If the soil is waterlogged, it might cut off the necessary oxygen supply and prevent the seed from germinating.

Temperature and light
Seeds germinate over a wide range of temperatures, with many preferring temperatures slightly higher than room-temperature. Often, seeds have a set temperature range for sprouting and will not sprout above or below a certain temperature. In addition, some seeds may require exposure to light or to cold temperature to break dormancy before they can germinate. As long as the seed is in its dormant state, it will not germinate even if conditions are favorable. For example, seeds requiring the cold of winter are inhibited from germinating if they never experience frost. Some seeds will only germinate when temperatures reach hundreds of degrees, as during a forest fire. Without fire, they are unable to crack their seed coats. Many seeds in forest settings will not germinate until an opening in the canopy allows then to receive sufficient light for the growing seedling.

Stratification

Seeds must be mature and environmental factors must be favorable before germination can take place. When a mature seed is placed under favorable conditions and fails to germinate, it is said to be dormant. Some seeds will not germinate (begin to grow) until they have been dormant for a while. The length of time plant seeds remain dormant can be reduced or eliminated by a simple seed treatment called stratification. Seeds should be planted promptly after stratification.

Stratification mimics natural processes that weaken the seed coat before germination. In nature, some seeds require particular conditions to germinate, such as the heat of a fire (e.g., many Australian native plants), or soaking in a body of water for a long period of time. Others have to be passed through an animal's digestive tract to weaken the seed coat and enable germination.

==9. Make a collection of 30 different kinds of seeds, of which only 10 may be collected from commercial seed packages, the other 20 you are to collect yourself. Label each kind as follows: seed name, date collected, location collected, and collector’s name. ==


text:
User talk:Klinkaeo~enwikibooks

Abstract The purpose this research were to compare with achievement students in Thai subject (Thai 31101)...E-mail : rungsun kk (AT) yahoo (DOT) com (klinkaeo (AT) gigya (DOT) com

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Renamed
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title: User talk:Klinkaeo~enwikibooks


text:
Croatian/Main Contents/Level 2/Lesson 2

Interrogative
Next Lesson

title: Croatian/Main Contents/Level 2/Lesson 2


text:
Horticulture/Rorippa sylvestris

Rorippa sylvestris (Creeping Yellowcress, Keek, Yellow Fieldcress; syn. Radicula sylvestris (L.) Druce ) is an invasive species ) in the United States.

Description
Rosette-forming perennial with pinnatifid leaves and clusters of yellow flowers. Colonizes rapidly via underground rhizomes.

Ecology
Considered a noxious weed in parts of the United States.

title: Horticulture/Rorippa sylvestris


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Cookbook:Traditional Finnish Lemon Mead (Sima)

| Beverages | Cuisine of Finland

Sima is a Finnish, lemon-flavored mead mostly consumed during the celebrations of the 1st of May. A glass of sima is traditionally accompanied by a funnel cake or sugar coated donut.

The purpose of the fermentation is mainly to make the drink bubbly; the alcohol content of traditionally prepared sima is only 0.5-0.8%. As the drink is meant to be consumed in reasonable quantities anyway, a glass of sima is generally considered suitable for children over five years of age.

Ingredients

 * 8 litres water
 * 500 g brown sugar
 * 500 g white sugar
 * 2–3 lemons
 * ¼ tsp yeast (ordinary baker's yeast, either fresh or dry)
 * 1 handful of raisins

Equipment

 * Container large enough to hold all of the sima (a big plastic bucket for example)
 * Empty bottles
 * Funnel

First day

 * 1) Sterilize all of your equipment. A thorough process is described in the mead recipe. However, as the fermentation time of sima is very short (only a few days) and it is usually consumed within a week or so, just washing the bucket and bottles well and then rinsing them with boiling water right before using is generally enough to prevent spoiling.
 * 2) Boil 4 litres of water.
 * 3) Wash the lemons well and peel the zest, being careful not to include the bitter white pith. Put the zest into your container.
 * 4) Slice the lemons and remove the white pith from around the slices. Add the slices into the container.
 * 5) Add all the sugar into the container with the lemons.
 * 6) Pour the boiling water on top, and mix well until the sugar is mostly dissolved.
 * 7) Add 4 litres of cold water so that the mixture will be lukewarm, the right temperature for the yeast to activate. If unsure, check with a thermometer—the temperature should be between 35°C and 45°C.
 * 8) When the temperature is right, add the yeast. One way is to take a small cup of the mixture from the container, dissolve the yeast in it and then mix it back. Just dropping a crumb of fresh yeast into the container will also work.
 * 9) Let it stand overnight in room temperature.

Second day

 * 1) Bottle it on the next day: add one spoonful of sugar and a few raisins to each bottle before you pour the sima in.
 * 2) After bottling, let the bottles stand at room temperature out of direct sunlight for 2 more days. The raisins will bloat up and float to the surface.
 * 3) Move the bottles to your refrigerator. The sima is ready, and should be served cold. It will keep for at least a week in the refrigerator.

Notes, tips and variations

 * For a simpler process, just squeeze in the juice of 1–2 lemons instead of removing the pith.
 * You may use 1 kg brown sugar instead of mixing white and brown to make a darker sima with a stronger brown sugar flavour.
 * All the sugar may optionally be replaced with 1 kg honey.
 * You can put the bottled sima into the refrigerator immediately instead of fermenting overnight at room temperature. This will cause the fermentation to be slower; the sima will be ready in approximately 5–7 days.
 * Plastic soda bottles are recommended as pressure can build up in the bottles. An exploding plastic bottle is less dangerous than an exploding glass bottle.


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Parkour/Exercises/The Pull-Up

The Pull-up is an upper-body compound exercise, involving many muscles. It requires a sturdy bar, above your shoulders but low enough for you to reach it and it must be able to support your body weight.

Steps

 * 1) Grasp the bar above your head firmly. Palms towards you is a chin-up (increasing the load on the biceps), while palms away is a traditional pull-up (increasing the load on the latissimus dorsi and other muscles of the back).  Employing a variety of hand positions from session to session or set to set is beneficial; even a "neutral" grip—the palms facing each other—can be used.  A neutral grip could be used, for example, if exercising between two parallel bars on a jungle gym.
 * 2) Curl your legs up so that they are not touching the ground if necessary.
 * 3) Use your arms to pull your whole body up.  Try and intentionally focus on the muscles being used during exertion for greater gains.  For example, when doing a traditional pull-up, as described above, focus on drawing the shoulder blades toward each other and down, trying to increase the "feel" of the muscles working.  When focusing on the movements this way, it becomes clear by sensation how differently the hand placement (i.e., towards or away from face) affects the musculature exercised during the movement.
 * 4) Pause when your chin is above the bar.
 * 5) Go back down slowly.
 * 6) Repeat as many times as necessary based on your exercise needs.  Some trainers advise that gains can be increased (or plateaus passed) if one continues on with this exercise after "failure."  Failure, as such, occurs when you can no longer pull yourself up to the bar.  To continue, you use the strength in your legs to actually jump up to the bar into a position as if you had already pulled yourself up, and slowly, in a controlled fashion, let yourself down again.  This makes the exercise an eccentric isotonic rather than the standard concentric isotonic, and allows one to exercise the involved muscles to a greater extent (and in a slightly different and physiologically valuable manner, see Eccentric contractions).

Tips

 * Practice. This is a difficult exercise and will take a long time to master. It is not a simple exercise and it requires a fair amount of strength.
 * Begin doing it slowly. Only engage in "ballistic" (i.e., fast moving, with explosive force) exercise if you are already practiced in the movement and know the effects and dangers of ballistic exercise. Nonetheless, swinging while performing the exercise is a waste—try and perform it in as controlled a manner as possible, regardless of the speed.  When exercise-training for applied sports, such as Parkour, which involve ballistic movements as part of their repertoire, it is important to train one's muscles with ballistic methods.  See Qualities and principles of ballistic training.

Variations

 * One handed pull-up: Use one hand instead of two, to pull yourself up.

title: Parkour/Exercises/The Pull-Up


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Wikijunior:Human Body/Anatomy

Anatomy describes the composition and relationships of parts of our body. Just describing how our body is composed of many parts helps us guess and memorize functions of our body. Anatomy is what this book is mostly about and it is fascinating to see how well our body is designed. Anatomy deals with organs or groups of organs. An organ is a distinct part of our body with a function, for example our heart. Anatomy of the movement apparatus (all organs that are involved in movement), for example, deals with muscles, bones and tendons. Groups of organs such like this are also called organ systems.

How to approach an organ?
If you learn about the anatomy of an organ or an organ system, you will soon notice that you can look at an organ from afar and also very close up. If you go to extremes you use a microscope to look at single cells of an organ. All organs are made from cells. Therefore we will understand a lot about an organ by both looking at it with the naked eye - which we will call the macroscopic view - and at cells - which we call the microscopic view. In the individual descriptions of organs you will therefore find both macroscopic and microscopic aspects of that organ.

We now list some subtopics of anatomy and where you can find more information about those organs in this book.

There are many different types of organs such as the heart, liver, brain etc.

Movement apparatus
The movement apparatus deals with the organs needed for movement: bones, muscles and tendons. While we often think of movement in terms of walking, climbing, or moving our mouths to talk, movement goes on inside of us as well. For example, our digestive systems use muscles in organs to transport food through the digestive process. Likewise, movement inside of our bodies by the diaphragm allows us to take in air from our surroundings.

Circulatory system
The so called circulatory system involves everything that keeps our blood in order and flowing. Organs are blood vessels, heart, bone marrow and the spleen. The liver also takes part in keeping our blood healthy, but it has many more functions, too. The lungs help in the purification and oxygenation of blood.

Respiratory system
The respiratory system allows us to breathe in order to take in oxygen. This involves the mouth, epiglottis, trachea and the lungs. The respiratory system closely interacts with the circulatory system, because one important function of the blood is to transport oxygen. The heart acts like a pump circulating all the clean blood to different parts of the body and pumps the impure blood for purification to the lungs. The main function of lung is exchange of gases i.e. transport of carbon dioxide and oxygen. There is a specialised network of pulmonary capillaries which carries out gases exchange across the alveoli. Oxygen enters pulmonary capillaries from alveoli and carbon dioxide enters into the alveoli from pulmonary capillaries. This exchange of gases occurs because of pressure gradient across the alveoli and pulmonary capillaries. Alveoli is the functional unit of lung.

Digestive system
Have you been hungry recently? Our digestive system deals with all the food we take in. Food travels from the mouth through the esophagus to the stomach. From there, the small intestine and the large intestine actually absorb nutrients (the small components in our food that our body can make use of) and water. Finally, faeces is excreted through the anus.

Urinary system
The urinary system we only feel when we urinate, or when we have an urge to do so. The kidneys are a filter that cleanses the blood. Also, if we drink more fluid than we need, excess fluid is transported through the kidneys as urine to the bladder. Since unhealthy substances are excreted through the urine, you can see that drinking enough fluid helps keep you healthy.

Reproductive system
The reproductive system differs between men and women. These organs are needed for a couple to have children. A baby is carried by a woman in the uterus. The special organ of the placenta is formed during pregnancy and is shared between mother and child who feeds through this organ.

Nervous system
The nervous system is composed of the brain, medulla and nerves. The nervous system contains our consciousness and we use our nervous system to control the rest of our body.

Other organs
The liver is a very important organ, producing many substances that other cells need. It plays an important part in collecting the digestive products with juices that aid digestion. It also stores energy (glycogen) which can be turned into quick form of energy (glucose). In this way the organ plays a major role in providing instant energy to supply important organs.

It is very important to remember that cutting off glucose and oxygen to the brain can make a person unconscious in 3 minutes. Thymus, spleen and tonsils in children help the immune system to defend against bacteria and viruses.

The largest organ on the human body is the epidermis, also known as our skin.

title: Wikijunior:Human Body/Anatomy


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User:Dr. K~enwikibooks

I made this account after a student telling me about Prentice Hall Chemistry, a book series I have found quite reliable in keeping up with new information.

Things to do

 * Category:User templates - I have found quite an interest in these userboxes that I have heard about. Since this Wikiproject is fairly new is there another Wikiproject with more userboxes? I will check Wikipedia first it seems that is the main project.

This is a test userbox that I have written in html code to practice

title: User:Dr. K~enwikibooks


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Horticulture/An Invitation to Contribute

Horticulture is meant to eventually become an exhaustive technical manual for professionals in the horticulture trade, as well as a guidebook for gardeners of every level of expertise. The scope of the book includes garden techniques, materials, soils, plants, weeds, diseases, pests, hotricultural chemicals, and beneficial insects.

Because of this large scope, this Wikimanual is not intended to be printed out as "one big book", since this would (after a few years of development) be far too thick a book to fit on any real-world bookshelf. Rather, it is meant to be printed out module-by-module, to cover topic areas for classrooms, or as "custom garden guides" for individual gardeners (only including those pages that pertain to a specific garden).

Because of the free licensing on Wikibooks, this means that any instructor can create textbooks for a specific class they are teaching, without any need to seek permissions or pay licensing fees. So, for example, an instructor could assemble a book for a two semester course on herbaceous ornamental plants that is specific to the lesson plan, and includes exhaustive discussion of how to grow and maintain any particular plant, from siting and fertilization to pest and disease control to commercial propagation methods.

Instructors can also use Wikibooks as a venue for class assignments: the edits of each student can be seen on their account logs, and they can of course be assigned particular tasks, such as writing entries for a set of plants, insects, diseases, and so on.

For amateur gardeners, this Wikimanual is a place for both learning and sharing "tips and tricks" they have learned from others or invented themselves. "Expert amateurs" often wish that "landscapers" knew how to do things better, and this Wikimanual is an excellent way to share (in fact, the free license allows anyone to print out copies and distribute them, and even tuck them under windshield wipers).

Ways to help
There are many ways in which to contribute, from indexing to copyediting to maintenance to creating new content. For cleanup and maintenance, a few templates and categories exist for this, such as Wmog-tw (for cleaning up transwikis that haven't yet been completely modified to fit the Wikimanual's Manual of Style).

Images are also needed for many pages, particularly those pages that describe a process such as ../Division/, ../Pruning/, and other garden techniques. These should be uploaded to Wikimedia Commons rather than to Wikibooks, because in the future much of this book might be translated into other languages.

There are also many stubs that need to be further developed.

Many pages are orphaned and need to be linked to from the table of contents or a listing linked to from the table of contents. See all the pages of the book in :.

New content is of course the most important need, since while there are already hundreds of "chapters", there are thousands of plants, pests, diseases, techniques, and so on. In general, the best way to start new chapters is by transwikiing content from Wikipedia using the Import tool, which copies all revisions and thus honors our copyrights. Only administrators can use the Import tool. To request an import from administrators, list the Wikipedia article name at WB:RFI.

/Project


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Cookbook:Peanut Sauce (Saus Kacang)

| Cuisine of Indonesia

Saus kacang is a spicy Indonesian peanut sauce.

Ingredients

 * 2 cans of natural peanut butter
 * Water
 * Sambal oelek
 * 1 lemon
 * Salt
 * Soy sauce

Procedure

 * 1) Mix the peanut butter with water, and stir gently until smooth.
 * 2) Add some sambal oelek, some lemon juice, salt, and soy sauce. Keep stirring, adjusting the ingredients to get the taste right.

Notes, tips, and variations

 * If using stabilized peanut butter, mix in a small amount of peanut or vegetable oil.

title: Cookbook:Peanut Sauce (Saus Kacang)


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Cookbook:Mixed Pickled Vegetables (Acar Campur)

| Cuisine of Indonesia

Acar campur is a very simple Indonesian pickle salad. Here is one example, of which there are many variations, some of which depend on what items are in season.

Ingredients

 * 1 fresh red pepper, seeded and sliced
 * 1 medium onions, quartered
 * 2 cloves garlic, crushed
 * ½-inch cube terasi (shrimp paste)
 * 4 macadamia nuts or 8 almonds
 * 1-inch piece of fresh turmeric or 1 teaspoon ground turmeric
 * ¼ cup corn, sunflower, or olive oil
 * 2 cups white vinegar
 * 1 cup water
 * ¼ cup white sugar
 * 3 medium carrots
 * 1 small cauliflower
 * 1 medium cucumber
 * 1 cup white cabbage
 * 1 cup dry-roasted peanuts, coarsely crushed
 * 1 teaspoon salt, to taste

Procedure

 * 1) Place the pepper, onion, garlic, shrimp paste, nuts, and turmeric in a food processor. Blend to a paste-like consistency.
 * 2) Heat the processed paste mixture in the oil for 2 minutes. Add the vinegar, water, sugar and salt. Bring to a boil while stirring. Simmer for 10 minutes, stirring occasionally, while preparing vegetables.
 * 3) Cut carrots into desired bite-sized shapes. Cut green beans into short, even lengths. Separate cauliflower into bite-sized florets. Peel seeded cucumber and cut into bite-sized cubes. Cut cabbage into bite-sized pieces.
 * 4) Blanch each vegetable separately in boiling water for 1 minute. Rinse in cold water to halt the cooking. Drain well.
 * 5) Add the vegetables to the boiled paste sauce. Gradually bring the entire mixture to a boil. Cook for 5–10 minutes. Do not overcook—the vegetables should still be crunchy.
 * 6) Remove from heat. Add the peanuts to the mixture and cool.
 * 7) Once cool, cover and chill for at least 3 days before serving.

Notes, tips and variations

 * Indonesian pickles tend to include a variety of shredded vegetables and spices, including the following:
 * Carrot
 * Cauliflower
 * Cucumber
 * Baby corn
 * Bean sprouts
 * Chili pepper
 * Ginger
 * Black pepper
 * Hazelnut
 * Pistachio
 * Pecan


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Wikijunior:Europe/Greece

Greece (officially the Hellenic Republic, Ελληνική Δημοκρατία in Greek) is a country in southern Europe. It shares borders with Bulgaria, North Macedonia, Albania and Turkey. The capital city is Athens. Other big cities in Greece are Thessaloniki, Patras and Heraklion. Greece is well known for its history and its islands. The country is also a member of the European Union and uses the Euro as its currency along with 16 other countries in Europe.

Greece's History
Greece is heir to the heritages of ancient Greece, the Roman and the Byzantine Empire, and nearly four hundred years of Ottoman rule. Greece is the birthplace of democracy, the Olympic Games, western books, and western drama including both tragedy and comedy. The Iliad, an epic poem written by Homer in the 8th Century BC, is the oldest work of western writing.

Ancient Greece was not a single country; Ancient Greece was divided into lots of small city-states in the south and two kingdoms in the north. Generally the south was much more advanced; Athens and Sparta, the most powerful Greek cities, were in the south. There were many wars between the cities, and it was not until 338 BC (338 'before Christ') when they were all united to fight against the Persians under King Philip II of Macedon, one of the two kingdoms of the north. When Philip died his son, Alexander The Great, took his place and campaigned against Persia, establishing a new era, the Hellenistic era.

After the Hellenistic era, in 146 BC, the Roman period began when the Romans ruled over Greek lands. After a century the whole Greek-speaking world became part of the Roman Empire. Greek was the language used by most of the people in the eastern parts of the Empire, while Latin was spoken more in the west. In 330 AD the capital of the Roman Empire changed; Byzantium, an ancient Greek city, became the new capital, which was then renamed to Constantinople (in Greek: 'The city of Constantine' meaning the emperor who chose Byzantium as the new capital). In the following years the Roman Empire was divided into two parts; the east part became the Byzantine Empire. In 1453 the siege of Constantinople by the Turks took place, the Ottoman Empire was then established.

For almost four centuries, Greece was part of the Ottoman Empire. On 25 March 1821 Greeks started a revolution so they could be independent. Greece became an independent country on 3 February 1830. Many wars occurred before Greece gained the lands it has today. Greece also took part in the two World Wars. Greece fought alongside the Allies in World War II and the Entente in World War I. A civil war took place after the World Wars, too. After a military junta of 1967–1974 a vote was held for Greeks to decide whether they wanted Greece to become a Republic or to remain a Kingdom. The Greek vote of December 8, 1974, resulted in the end of the monarchy in Greece and the establishment of the current Third Hellenic Republic.

Greece became the tenth member of the European Union on January 1, 1981. The country adopted the Euro in 2001 and successfully organized the 2004 Olympic Games in Athens. At the moment Greece is facing big economic and social problems.

Greece's Geography
Four fifths of Greece consist of mountains or hills, making the country one of the most mountainous in Europe. The highest mountain in Greece is Olympus, the mythical house of the ancient Gods. Western Greece contains a number of lakes and wetlands. Greece has also many islands, over 6,000, the largest being Crete. Other well-known large islands are Rhodes, Santorini and Corfu.

Greece is an active earthquake zone, small earthquakes occur every year in Greece, especially in Western and Southern Greece. Greece has a coastline of over 13,600 kilometres. Mount Olympus, at 2,917 metres, is the highest mountain in the country and is believed to be the home of the Greeks Gods in Greek Mythology.

The climate of Greece can be categorized into three types: the Mediterranean, the Alpine and the Temperate. The Mediterranean type of climate features mild, wet winters and hot, dry summers. The Alpine type of climate features harsh winters with large snowfalls, while the summers are cool with many thunderstorms. This climate is to be found on high mountains. The Temperate type of climate features cold, damp winters and hot, dry summers.

Greece's People
Greece's population is around 11 million. The majority of people living in the country are ethnic Greeks with some people from foreign countries like Albania and Bulgaria staying in the country. The most spoken language in Greece is Greek, as well as the official language of the state. But there are also people who speak Bulgarian and Turkish close to the borders with Bulgaria and Turkey. Greek language has a history of almost 4,000 years. English is widely spoken as a foreign language, along with other languages such as French, German, Italian and Spanish among others. These five languages are also provided around the country's schools. Education in Greece is required from the ages of 5 to 15.

The majority of Greek people are Greek (Eastern) Orthodox (95%) and it is also the state religion. There are other Christian groups in Greece, too, the largest being Catholic Christians; the majority lives in the Cyclades and the Ionian Islands. The country's oldest religious minority is the Jewish community, however, it is no longer the largest non-Christian group. There are also some Muslims residing in the Northern Greek region of Thrace, close to the Turkish borders.

Greek cuisine is a typical Mediterranean cuisine, similar to those of Italy, the Balkans, Turkey, and the Middle East. The majority of national foods are based around olive oil, vegetables and herbs, grains and bread, wine, fish, and various meats. Also important are olives, cheese, aubergine, courgette, and yoghurt. Greek desserts are characterized by the dominant use of nuts and honey. Greek salad is widely known around the globe.

The most famous sport among Greeks is probably football. Greeks also tend to listen to foreign music a lot, while Greek music is famous in countries in the Balkans and the Middle East.

Greece's Sights
Greece attracts more than 16 million tourists each year. Greece has been an attraction for international visitors for a long time because of its rich and long history and more recently for its glorious Mediterranean coastline and beaches. The most popular destinations for tourists are the Cyclades, Athens, Crete, Rhodes, the Ionian Islands, where the island of Corfu is, and other coastal regions of the country as well as places of archaeological interest.

There are 17 sites that are included in the list of UNESCO World Heritage Sites, Among them are the Acropolis of Athens, the ancient sites of Vergina and Olympia but also the old cities of Corfu and Rhodes.

title: Wikijunior:Europe/Greece


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Wikijunior:Europe/Albania

Albania (Shqipëri in Albanian) is a small country in south-eastern Europe in the Balkan peninsula. It shares borders with Greece, Montenegro, Kosovo, and Macedonia. The capital is Tirana. The currency of Albania is the Lek. Other large cities are Durrës and Elbasan. Albania isn't part of the European Union.

Albania's History
During the fifteenth century Albania enjoyed a brief period of independence under the legendary hero, Skanderbeg. Aside from this exception, the country did not enjoy independence until the twentieth century. After five hundred years of Ottoman rule, an independent Albania was proclaimed in 1912. The country adopted a republican form of government in 1920. In 1939 the Italians invaded the country. Because of this event Albania was one of the first countries occupied by the Axis Powers in World War II.

After the war Albania became a communist state known as the Socialist People's Republic of Albania and was ruled by Enver Hoxha until 1985. Albania became a democratic country in 1991 but faced problems throughout the 1990s. In 1996 there were severe economic problems because of the failure of Ponzi schemes in the country. In 1997 there was an armed rebellion and hundreds of thousands of people emigrated to other parts of Europe and North America. In 1999 Albania faced an influx of refugees from Kosovo, Serbia during violence in that province.

Albania's Geography
Albania has a total area of 28,750 square kilometers. Albania has coastline on the Adriatic Sea and the Ionian Sea. Albania has a continental climate with hot summers and fairly cold temperatures. 70% of the country is mountainous with the highest mountain being Korab at 2,753 metres.

Albania has many fairly big lakes including one of central Europe's biggest lakes - Lake Shkodër - which is located partly in Albania and partly in neighbouring Montenegro.

Albania's People
Albania has been populated since prehistoric times and in recent history the country has been home to large numbers of Catholic Christians and Jews but in the past few hundred years people have either converted to Islam or moved elsewhere. About 40% of the population are Muslim, 40% are Christian and 20% are Atheist (no religion). One of the most internationally famous Albanians was Agnes Gonxha Bojaxhiu, better known as Mother Teresa who was a Roman Catholic nun who worked with the poor around the world, especially India. Between 1945 and 1992 Albania was a communist country but currently the country is hoping to join organizations like the European Union. The national language is Albanian. Many Albanians also speak English, Italian and Greek. A large number of Albanians also live in Macedonia, Greece, Italy and in Kosovo.

The literacy rate for the total population, age 9 or older, is about 94%. Elementary education is required for grades 1-9, but most students continue at least until a secondary education (grades 10-12). Students must successfully pass graduation exams at the end of the 9th grade and at the end of the 12th grade in order to continue their education. The school year begins in September and finishes in late May or early June.

The most popular sport in Albania is football and the most followed sports event is the World Cup. Other played sports include basketball, volleyball, and gymnastics.

Albania's Sights
Today Albania is a popular destination for tourists from Europe and elsewhere. Tourists visit for the Greek, Roman and Ottoman archaeological buildings as well as some remnants from Communist times. The mountains and beaches in Albania are said to be as good, or better, than elsewhere in Europe but much less crowded. In 2011 Lonely Planet said that Albania is the number 1 destination to visit and most tourists who visit come from Kosovo (Serbia), Greece, Macedonia and Italy. Other visitors come from Poland, Czech Republic, France and Sweden.

There are 3 UNESCO World Heritage Sites in Albania including the Greek and Roman city of Butrint and a medieval Ottoman town called Gjirokastër.

title: Wikijunior:Europe/Albania


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Wikijunior:Europe/Lithuania

Lithuania is a country in north eastern Europe. It shares borders with Poland, Russia, Belarus, and Latvia. The capital city is Vilnius. Other big cities in Lithuania are Kaunas and Klaipėda. Lithuania has been part of the European Union since 2004. Lithuania's currency is the Euro

Lithuania's History
Lithuania was first formed in the 6th century. Lithuania first appeared on the map in the middle of 13th century when Mindaugas united Lithuanian lands. He became a King in 1253 and then Lithuania was recognised as the Kingdom of Lithuania. Later on, the country expanded quickly and in the beginning of the 15th century Lithuanian Duchy was one of the largest countries in Europe, stretching from Baltic to the Black sea. However, due to a great threat of being occupied by Grand Principality of Moscow, Lithuania united with Poland in 1569 into a Polish-Lithuanian Commonwealth country. Lithuania had its own army, currency, laws (the three statutes of Lithuania) and institutions.

In 1795 during the third Partition of the Commonwealth, the country was occupied by Russia, Prussia and Austria. Majority of Lithuanian lands were in the Russian Empire with the rest of it in Prussia. Lithuania re-established its independence after the First World War on February 16, 1918. During the Second World War, The Soviet Union occupied Lithuania in June 1940, one year later Lithuania was occupied by German forces and was reoccupied by Soviet Union in 1944. More than 13% of Lithuanian population, including Lithuanian Jews, were killed during the war, many of them by the secret Soviet police forces were deported to Siberian GULAGs where people died from cold and hunger, the others had been sent to German labour camps.

March 11, 1990 Lithuania was the first Soviet Republic to renew its independence. Lithuania joined the United Nations on September 17, 1991 and on May 31, 2001 it became a member of the World Trade Organization (WTO). On January 4, 1994, it became the first of the Baltic states to apply for NATO membership. On March 29, 2004, it became a NATO member, and on May 1, 2004, Lithuania joined the European Union. Lithuania adopted the Euro as its currency in 2015.

Lithuania's Geography
Lithuania is situated in northern Europe. It has sandy coastline with big dunes some of which face the open Baltic Sea. There are no mountains in Lithuania, only hills which are not higher that 300 metres (1,000ft). However, there are many lakes and wetlands. The main river is Neman (Nemunas). Green forests cover about 30% of the land. Climate is wet and moderate, though sometimes weather fluctuations happen with -25C in winter and 38C in summer. According to one geographical computation method, Lithuania's capital, Vilnius, lies only a few kilometres south of the geographical centre of Europe.

Lithuania's People
There are only 3.3662 million people living in Lithuania, 84.6% of whom are ethnic Lithuanians who speak the Lithuanian language, the official language of the country. Together with Latvian, it is one of the two surviving members of the Baltic language group and one of the oldest used spoken languages in the entire world. There are also some minorities in Lithuania, including Poles (6.3%), Russians (5.1%), and Belarusians (1.1%).

Most Lithuanian schools teach English as a first foreign language, but students may also study German, or, in some schools, French. The majority of Lithuanians are Christian Catholics, even though they were very last in Europe to be baptised. It is said that many Lithuanians were following Baltic pagan religion until the middle of 18th century. Many traditions, celebrations and songs come from the pagan times. People then believed in many gods (gods of earth, wind, thunder, water, etc.).

Since 2004 many Lithuanians have moved to other parts of the EU, especially Germany, the UK and Ireland, to work and live.

Lithuania's Sights
Today Lithuania is a popular tourist destination. Most tourists come from the neighbouring Baltic countries, Russia, Germany, Poland and the UK. Vilnius, the capital, is a popular city for cultural museums and night-life. There are three UNESCO World Heritage Sites in Lithuania: Kernavė (an archaeological site), Curonian Spit (shared with Russia) and the Old Town of Vilnius. The Hill of Crosses is a notable religious tourist attraction.

title: Wikijunior:Europe/Lithuania


text:
Wikijunior:Europe/Iceland

id:Wikijunior:Eropa/Islandia

Iceland is an island country in the far north west of Europe and is famous for its natures beauty and variety also the loftiness Icelandic Sagas who tell the tales of the Vikings and their survivors. Iceland does not border any other country and the capital city is Reykjavik where almost 70% of the Icelandic population live. Iceland is not part of the European Union. The Krona is the currency of Iceland.

Iceland's History

It is generally believed that Iceland was discovered and settled by Norse Vikings in the second half of the 9th. The first settler was Ingólfur Arnason who came to Iceland with his brother Hjörleifur. Hjörleifur was killed by his slaves but Ingólfur built his homestead in Reykjavík in 874. From 874 to 930 lots of people, mostly Norse Vikings and their Irish slaves moved to the island and by 930, most available farmland had been claimed and the Althing, a parliament was founded as the political hub of the Icelandic Free State. Christianity was peacefully adopted in the year 1000.

In 1814, following the Napoleonic Wars, Denmark-Norway was broken up into two separate kingdoms by the way of the Treaty of Kiel. Iceland, however, remained a Danish dependency. The country's climate worsened during the 19th century, leading to mass emigrations to North America, largely Canada. Meanwhile, a new independence movement arose under the leadership of Jón Sigurðsson, inspired by the romantic and nationalist ideologies of mainland Europe. In 1874, Denmark granted Iceland home rule, which was expanded in 1904. In 1918, Iceland became fully independent but the head of state remained as the King of Denmark.

During World War II, the German occupation of Denmark on April 9, 1940 severed communications between Iceland and Denmark. A month later, British military forces sailed into Reykjavík harbor, violating Icelandic neutrality. Allied occupation of Iceland lasted throughout the war. In 1941, responsibility for the occupation was taken over by the United States Army. Following a national vote on the subject, Iceland formally became an independent republic on June 17, 1944. The occupation force left in 1946. Iceland became a member of NATO on March 30, 1949, amid domestic controversy and riots. On May 5, 1951, a defence agreement was signed with the United States -- American troops returned and stayed as part of the defence agreement throughout the Cold War and until the autumn of 2006.

Since World War II Iceland's economy has grown incredibly. The most common industries are financial services, tourism and fishing.

Iceland's Geography
Iceland is an island with no borders. It is often referred to as the island of ice and fire and is constantly being shaped by Glaciers and volcanoes. The island is about 103,000 km2 (39,768.5 sq mi) meaning that per 3 Icelanders has 1 km2 for them self. About 23,805 km2 (9,191 sq mi) of the island is Vegetation, 2,757 km2 (1,065 sq mi) are Lakes and about 11,922 km2 (4,603 sq miles) are Glaciers. Iceland is geologically a young land, Iceland is located on a geological so called hot spot, thought to be caused by a mantle plume. Meaning that under the island is a hot red burning lava coming up all the way from the inner earth The island is also over the Mid-Atlantic Ridge, which runs right through it. This combined location means that geologically the island is extremely active, having many volcanoes, notably Hekla, Eldgjá, and Eldfell. The volcanic eruption of Laki in 1783–1784 caused a famine that killed nearly a quarter of the island's population; the eruption caused dust clouds and haze to appear over most of Europe and parts of Asia and Africa for several months after the eruption. Iceland has hundreds of mountains and volcanoes. The highest mountain is Hvannadalshnúkur at 2,110 metres.

Because Iceland is over a so called hot spot there are lots of hot springs and are also geysers. A word derived from the name of the most famous one in Iceland, Geysir in Haukadalur the most geologically active is Strokkur in the same area. With such a widespread availability of geothermal power, and also because of the numerous rivers and waterfalls that are harnessed for hydro power, residents of most towns have hot water and home heat for a low price. The island itself is composed primarily of basalt, a low-silica lava associated with effusive volcanism like Hawaii. There are, however, a variety of volcano-types on Iceland that produce more evolved lavas.

Iceland's People
The Icelandic Population is about 300.000 people which is about the size of a small city in Europe therefore there are lots of things that make the Icelandic people so special for one their language.

Icelanders write and speak Icelandic, a North Germanic language descended from Old Norse. It is the only living language to retain the runic letter Þ (pronounced "TH") which indicates the fact that Icelandic has changed less from Old Norse than the other Nordic languages and has a considerable extent development of new vocabulary based on native roots rather than borrowings from English. Because of that Icelanders can still read and understand Old Norse which their famous sagas are written in. The Sagas were written around the time of the island's settlement. Until the Christianization of Iceland, many traditional Viking beliefs held strong, remnants of which remain today. For example, some Icelanders either believe in elves or are unwilling to rule out their existence. Christianity is the main religion in Iceland.

The majority of national foods are based around fish, lamb and dairy products. Þorramatur is a national food, consisting of many different dishes; this is not consumed on a daily basis but usually around the month of Þorri. Traditional dishes include skyr, cured ram scrota, cured shark, singed sheep heads and black pudding.

Though changing in the past years, Icelanders remain a very healthy nation. Children and teenagers participate in various types of leisure activities. Popular sports today are mainly football, track and field, handball and basketball. Sports such as golf, tennis, swimming, chess and horseback riding on Icelandic horses are also popular.

Iceland is home to the television station Nick Jr.'s animated program LazyTown (Icelandic: Latibær), a television program created by Magnús Scheving. It has become a very popular program for children and adults, and is shown in over 98 countries, including the US, Canada, Sweden and Latin America.[17] The LazyTown Studios are located in Garðabær. Iceland was also the home of The Sugarcubes, a popular 80's and 90's pop group; whose members included the now famous singer Björk.

Iceland's Sights
500,000 tourists visit Iceland every year. Popular attractions include the rugged landscape of Iceland which is generally peaceful because there are so few people in the country. Volcanoes, mountains and lakes are all extremely popular because of their sheer number. The capital city is popular because of its small city when compared to other world capitals. Iceland has 2 UNESCO World Heritage Sites: Þingvellir National Park and Surtsey, which is a small volcanic island of the Icelandic coast.

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Wikijunior:Europe/Norway

Norway is a country in north-western Europe. It shares borders with Sweden, Finland, and Russia. The capital city is Oslo. Other big cities in Norway are Bergen, Stavanger, Trondheim and Kristiansand. Norway's currency is called the Norwegian krone. Norway is not a member of the European Union. The Norwegian people voted "no" for joining the EU. id:Wikijunior:Eropa/Norwegia

Norway's History
Norway's history is dominated by the sea, which has provided much of the wealth and income to the expert seafarers who lived here. From 900-1349 A.D, Norway is marked by war as warlords try to conquer each other's lands and as Vikings from Norway travel the seas in search of wealth and land. With their superior boats, they traveled on shallow rivers and pillaged places which were thought inaccessable, but also traded wares all over europe. These Vikings discovered and settled far-off places like Newfoundland and Iceland and they conquered places like Normandy and the Orkneys.

After the Black Death in 1350, Norway lost half of its population and it became part of The Kalmar Union (Norway, Denmark and Sweden) during 1397-1450. When Sweden went out of the union, Norway was a province owned by Denmark from 1450-1814. In 1814, Norway made its own constitution. Because of the Napoleonic wars, Norway was made mostly autonomous, albeit in an union with Sweden. A feeling of nationalism and conflicting trade policies made the union unpopular in Norway, and by 1905 Norway became fully independent from Sweden without a war. After an election, the Norwegian people chose to be a democracy with a king, and they found the Danish King's son for the role. He took the more Norwegian-like name Haakon, and his son took the name Olav. The Queen continued to be Maud.

Norway's Geography
Fjords, long inlets from the sea, cover most of Norway's coastline. Norway is a very long country, like Chile. It stretches from 57° and 81° North and parts of the north are in the Arctic Circle.

Norway has 239,057 registered islands spread across the coastline of 83,000 kilometres.

Norway's highest mountain is Galdhøpiggen at 2,469 metres.

Norway's People
Norways' population is mostly ethnic Norwegians and people from the other Scandinavian countries. Approximately 5 million people live in Norway.

The majority of Norwegians are Protestant Christian. There are also small minorities of Catholic Christians, Muslims and Jews.

The main language of Norway is Norwegian (which has 2 written forms). A variety of Sami languages are spoken by the Sami people especially in the north of the country.

One famous Norwegian artist was Edvard Munch who is famous for The Scream. Most of the famous artists such as Henrik Ibsen or Edvard Grieg lived during the 1800s.

Norway's Sights
Norway is very popular with tourists especially those who like the great outdoors. Norway is a very expensive country but many tourists visit every year to see the famous fjords, the unspoiled nature and the attractive towns around the country.

The most visited tourist sites in Norway are Holmenkollen Ski Jump and Bryggen which is a UNESCO World Heritage Site. The capital, Oslo, is popular for business trips and sight-seeing trips and cruises in the Arctic north are popular especially during the winter.

title: Wikijunior:Europe/Norway