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Linear span

In mathematics, the linear span (also called the linear hull or just span) of a set $S$ of vectors (from a vector space), denoted $span(S)$, is defined as the set of all linear combinations of the vectors in $S$. For example, two linearly independent vectors span a plane. The linear span can be characterized either as the intersection of all linear subspaces that contain $S$, or as the smallest subspace containing $S$. The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules.

To express that a vector space $V$ is a linear span of a subset $S$, one commonly uses the following phrases—either: $S$ spans $V$, $S$ is a spanning set of $V$, $V$ is spanned/generated by $S$, or $S$ is a generator or generator set of $V$.

Definition
Given a vector space $V$ over a field $K$, the span of a set $S$ of vectors (not necessarily finite) is defined to be the intersection $W$ of all subspaces of $V$ that contain $S$. $W$ is referred to as the subspace spanned by $S$, or by the vectors in $S$. Conversely, $S$ is called a spanning set of $W$, and we say that $S$ spans $W$.

Alternatively, the span of $S$ may be defined as the set of all finite linear combinations of elements (vectors) of $S$, which follows from the above definition.

$$ \operatorname{span}(S) = \left \{ {\left.\sum_{i=1}^k \lambda_i \mathbf v_i \;\right|\; k \in \N, \mathbf v_i  \in S, \lambda _i  \in K} \right \}.$$

In the case of infinite $S$, infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming that such sums are defined somehow as in, say, a Banach space) are excluded by the definition; a generalization that allows these is not equivalent.

Examples
The real vector space $$\mathbb R^3$$ has {(−1, 0, 0), (0, 1, 0), (0, 0, 1)} as a spanning set. This particular spanning set is also a basis. If (−1, 0, 0) were replaced by (1, 0, 0), it would also form the canonical basis of $$\mathbb R^3$$.

Another spanning set for the same space is given by {(1, 2, 3), (0, 1, 2), (−1, $1/2$, 3), (1, 1, 1)}, but this set is not a basis, because it is linearly dependent.

The set ${(1, 0, 0), (0, 1, 0), (1, 1, 0)}$ is not a spanning set of $$\mathbb R^3$$, since its span is the space of all vectors in $$\mathbb R^3$$ whose last component is zero. That space is also spanned by the set {(1, 0, 0), (0, 1, 0)}, as (1, 1, 0) is a linear combination of (1, 0, 0) and (0, 1, 0). Thus, the spanned space is not $$\mathbb R^3.$$ It can be identified with $$\mathbb R^2$$ by removing the third components equal to zero.

The empty set is a spanning set of {(0, 0, 0)}, since the empty set is a subset of all possible vector spaces in $$\mathbb R^3$$, and {(0, 0, 0)} is the intersection of all of these vector spaces.

The set of monomials $x^{n}$, where $n$ is a non-negative integer, spans the space of polynomials.

Equivalence of definitions
The set of all linear combinations of a subset $S$ of $V$, a vector space over $K$, is the smallest linear subspace of $V$ containing $S$.

 * Proof. We first prove that $span S$ is a subspace of $V$. Since $S$ is a subset of $V$, we only need to prove the existence of a zero vector $0$ in $span S$, that $span S$ is closed under addition, and that $span S$ is closed under scalar multiplication. Letting $$S = \{ \mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n \}$$, it is trivial that the zero vector of $V$ exists in $span S$, since $$\mathbf 0 = 0 \mathbf v_1 + 0 \mathbf v_2 + \cdots + 0 \mathbf v_n$$. Adding together two linear combinations of $S$ also produces a linear combination of $S$: $$(\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n) + (\mu_1 \mathbf v_1 + \cdots + \mu_n \mathbf v_n) = (\lambda_1 + \mu_1) \mathbf v_1 + \cdots + (\lambda_n + \mu_n) \mathbf v_n$$, where all $$\lambda_i, \mu_i \in K$$, and multiplying a linear combination of $S$ by a scalar $$c \in K$$ will produce another linear combination of $S$: $$c(\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n) = c\lambda_1 \mathbf v_1 + \cdots + c\lambda_n \mathbf v_n$$. Thus $span S$ is a subspace of $V$.

 * Suppose that $W$ is a linear subspace of $V$ containing $S$. It follows that $$S \subseteq \operatorname{span} S$$, since every $v_{i}$ is a linear combination of $S$ (trivially). Since $W$ is closed under addition and scalar multiplication, then every linear combination $$\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n$$ must be contained in $W$. Thus, $span S$ is contained in every subspace of $V$ containing $S$, and the intersection of all such subspaces, or the smallest such subspace, is equal to the set of all linear combinations of $S$.

Size of spanning set is at least size of linearly independent set
Every spanning set $S$ of a vector space $V$ must contain at least as many elements as any linearly independent set of vectors from $V$.

 * Proof. Let $$S = \{ \mathbf v_1, \ldots, \mathbf v_m \}$$ be a spanning set and $$W = \{ \mathbf w_1, \ldots, \mathbf w_n \}$$ be a linearly independent set of vectors from $V$. We want to show that $$m \geq n$$.

 * Since $S$ spans $V$, then $$S \cup \{ \mathbf w_1 \}$$ must also span $V$, and $$\mathbf w_1$$ must be a linear combination of $S$. Thus $$S \cup \{ \mathbf w_1 \}$$ is linearly dependent, and we can remove one vector from $S$ that is a linear combination of the other elements. This vector cannot be any of the $w_{i}$, since $W$ is linearly independent. The resulting set is $$\{ \mathbf w_1, \mathbf v_1, \ldots, \mathbf v_{i-1}, \mathbf v_{i+1}, \ldots, \mathbf v_m \}$$, which is a spanning set of $V$. We repeat this step $n$ times, where the resulting set after the $p$th step is the union of $$\{ \mathbf w_1, \ldots, \mathbf w_p \}$$ and $m - p$ vectors of $S$.

 * It is ensured until the $n$th step that there will always be some $v_{i}$ to remove out of $S$ for every adjoint of $v$, and thus there are at least as many $v_{i}$'s as there are $w_{i}$'s—i.e. $$m \geq n$$. To verify this, we assume by way of contradiction that $$m < n$$. Then, at the $m$th step, we have the set $$\{ \mathbf w_1, \ldots, \mathbf w_m \}$$ and we can adjoin another vector $$\mathbf w_{m+1}$$. But, since $$\{ \mathbf w_1, \ldots, \mathbf w_m \}$$ is a spanning set of $V$, $$\mathbf w_{m+1}$$ is a linear combination of $$\{ \mathbf w_1, \ldots, \mathbf w_m \}$$. This is a contradiction, since $W$ is linearly independent.

Spanning set can be reduced to a basis
Let $V$ be a finite-dimensional vector space. Any set of vectors that spans $V$ can be reduced to a basis for $V$, by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the axiom of choice holds, this is true without the assumption that $V$ has finite dimension. This also indicates that a basis is a minimal spanning set when $V$ is finite-dimensional.

Generalizations
Generalizing the definition of the span of points in space, a subset $X$ of the ground set of a matroid is called a spanning set if the rank of $X$ equals the rank of the entire ground set

The vector space definition can also be generalized to modules. Given an $R$-module $A$ and a collection of elements $a_{1}$, ..., $a_{n}$ of $A$, the submodule of $A$ spanned by $a_{1}$, ..., $a_{n}$ is the sum of cyclic modules $$Ra_1 + \cdots + Ra_n = \left\{ \sum_{k=1}^n r_k a_k \bigg| r_k \in R \right\}$$ consisting of all R-linear combinations of the elements $a_{i}$. As with the case of vector spaces, the submodule of A spanned by any subset of A is the intersection of all submodules containing that subset.

Closed linear span (functional analysis)
In functional analysis, a closed linear span of a set of vectors is the minimal closed set which contains the linear span of that set.

Suppose that $X$ is a normed vector space and let $E$ be any non-empty subset of $X$. The closed linear span of $E$, denoted by $$\overline{\operatorname{Sp}}(E)$$ or $$\overline{\operatorname{Span}}(E)$$, is the intersection of all the closed linear subspaces of $X$ which contain $E$.

One mathematical formulation of this is

 * $$\overline{\operatorname{Sp}}(E) = \{u\in X | \forall\varepsilon > 0\,\exists x\in\operatorname{Sp}(E) : \|x - u\|<\varepsilon\}.$$

The closed linear span of the set of functions xn on the interval [0, 1], where n is a non-negative integer, depends on the norm used. If the L2 norm is used, then the closed linear span is the Hilbert space of square-integrable functions on the interval. But if the maximum norm is used, the closed linear span will be the space of continuous functions on the interval. In either case, the closed linear span contains functions that are not polynomials, and so are not in the linear span itself. However, the cardinality of the set of functions in the closed linear span is the cardinality of the continuum, which is the same cardinality as for the set of polynomials.

A useful lemma
Let $X$ be a normed space and let $E$ be any non-empty subset of $X$. Then 1. $\overline{\operatorname{Sp}}(E)$ is a closed linear subspace of X which contains E, | $\overline{\operatorname{Sp}}(E) = \overline{\operatorname{Sp}(E)}$, viz. $\overline{\operatorname{Sp}}(E)$ is the closure of $\operatorname{Sp}(E)$, | $E^\perp = (\operatorname{Sp}(E))^\perp = \left(\overline{\operatorname{Sp}(E)}\right)^\perp.$
 * undefined

(So the usual way to find the closed linear span is to find the linear span first, and then the closure of that linear span.)

Textbooks

 * Lay, David C. (2021) Linear Algebra and Its Applications (6th Edition). Pearson.
 * Lay, David C. (2021) Linear Algebra and Its Applications (6th Edition). Pearson.
 * Lay, David C. (2021) Linear Algebra and Its Applications (6th Edition). Pearson.
 * Lay, David C. (2021) Linear Algebra and Its Applications (6th Edition). Pearson.
 * Lay, David C. (2021) Linear Algebra and Its Applications (6th Edition). Pearson.
 * Lay, David C. (2021) Linear Algebra and Its Applications (6th Edition). Pearson.
 * Lay, David C. (2021) Linear Algebra and Its Applications (6th Edition). Pearson.

title: Linear span


text:
Gerald of Wales

Gerald of Wales (Giraldus Cambrensis; Gerallt Cymro; Gerald de Barri; c. 1146) was a Cambro-Norman priest and historian. As a royal clerk to the king and two archbishops, he travelled widely and wrote extensively. He studied and taught in France and visited Rome several times, meeting the Pope. He was nominated for several bishoprics but turned them down in the hope of becoming Bishop of St Davids, but was unsuccessful despite considerable support. His final post was as Archdeacon of Brecon, from which he retired to academic study for the remainder of his life. Much of his writing survives.

Early life
Born c. 1146 at Manorbier Castle in Pembrokeshire, Wales, Gerald was of mixed Norman and Welsh descent. Gerald was the youngest son of William Fitz Odo de Barry (or Barri), the common ancestor of the De Barry family of Ireland, a retainer of Arnulf de Montgomery and Gerald de Windsor, and one of the most powerful Anglo-Norman barons in Wales. His mother was Angharad FitzGerald, a daughter of Gerald FitzWalter of Windsor, Constable of Pembroke Castle, and his wife Nest ferch Rhys, daughter of Rhys ap Tewdwr, the last King of South Wales. Through his mother Angharad, Gerald was a nephew of David FitzGerald, Bishop of St Davids, as well as a great-nephew of Gruffydd ap Rhys, the son and heir of Rhys ap Tewdwr, and a cousin of Rhys ap Gruffydd, the famous Arglwydd (Lord) Rhys and his family.

Gerald received his initial education at the Benedictine house of Gloucester, followed by a period of study in Paris from c. 1165–74, where he studied the trivium. He was employed by Richard of Dover, the Archbishop of Canterbury, on various ecclesiastical missions in Wales, and distinguished himself by his efforts to remove supposed abuses of consanguinity and tax laws flourishing in the Welsh church at the time. He was appointed in 1174 as Archdeacon of Brecon, to which was attached a residence at Llanddew. He obtained this position by reporting the existence of the previous archdeacon's mistress; the man was promptly dismissed. While administering this post, Gerald collected tithes of wool and cheese from the populace; the income from the archdeaconry supported him for many years.

Upon the death of his uncle, the Bishop of St Davids, in 1176, the chapter nominated Gerald as his successor. St Davids had the long-term aim of becoming independent of Canterbury, and the chapter may have thought that Gerald was the man to take up its cause. King Henry II of England, fresh from his struggle with Archbishop Thomas Becket, promptly rejected Gerald in favour of Peter de Leia, one of his Norman retainers, possibly because Gerald's Welsh blood and ties to the ruling family of Deheubarth made him seem like a troublesome prospect. According to Gerald, the king said at the time: "It is neither necessary nor expedient for king or archbishop that a man of great honesty or vigour should become Bishop of St Davids, for fear that the Crown and Canterbury should suffer thereby. Such an appointment would only give strength to the Welsh and increase their pride." The chapter acquiesced in the decision; and Gerald, disappointed with the result, withdrew to the University of Paris. From c. 1179-8, he studied and taught canon law and theology. He returned to England and spent an additional five years studying theology. In 1180, he received a minor appointment from the Bishop of St Davids, which he soon resigned.

Travels
Gerald became a royal clerk and chaplain to King Henry II of England in 1184, first acting as mediator between the crown and Prince Rhys ap Gruffydd. He was chosen to accompany one of the king's sons, John, in 1185 on John's first expedition to Ireland. This was the catalyst for his literary career; his work Topographia Hibernica (first circulated in manuscript in 1188, and revised at least four times) is an account of his journey to Ireland; Gerald always referred to it as his Topography, though "history" is the more accurate term.

He followed it up, shortly afterwards, with an account of Henry's conquest of Ireland, the Expugnatio Hibernica. Both works were revised and added to several times before his death, and display a notable degree of Latin learning, as well as a great deal of prejudice against foreign people. Gerald was proud to be related to some of the Norman invaders of Ireland, such as his maternal uncle Robert FitzStephen and Raymond FitzGerald, and his influential account, which portrays the Irish as barbaric savages, gives important insight into Cambro-Norman views of Ireland and the history of the invasion.

Having thus demonstrated his usefulness, Gerald was selected to accompany the Archbishop of Canterbury, Baldwin of Forde, on a tour of Wales in 1188, the object being a recruitment campaign for the Third Crusade. His account of that journey, the Itinerarium Cambriae (1191) was followed by the Descriptio Cambriae in 1194.

His two works on Wales remain very valuable historical documents, useful for their descriptions (however untrustworthy and inflected by ideology, whimsy, and his unique style) of Welsh and Norman culture. It is uncertain whether Gerald was a Welsh speaker; although he quotes Welsh proverbs and appears familiar with the language, he seems not to have been employed as an interpreter for the expedition.

As a royal clerk, Gerald observed significant political events first-hand and was offered appointments as bishoprics of Wexford and Leighlin, and apparently, slightly later, the bishopric of Ossory and the archbishopric of Cashel, and later the bishopric of Bangor in Wales; and, in 1191, that of Llandaff. He turned them all down, possibly in the hope of landing a more prominent bishopric in the future. He was acquainted with Walter Map, whose career shares some similarities with Gerald's. Retiring from royal service, he lived in Lincoln from c. 1196 to 1198, when his friend, William de Montibus, was chancellor of the cathedral. In this period De principis instructione was probably first written, a useful historical source on contemporary events. It was an influential work at the time, spreading, for example, the legend of MacAlpin's treason. Here Gerald is frequently critical of the rule of the Angevin kings, a shift from his earlier praise of Henry II in the Topographia. He also wrote a life of St Hugh of Lincoln.

Attempts to become bishop of St Davids
On the death of Peter de Leia in 1198, the chapter of St Davids again nominated Gerald for the bishopric; but Hubert Walter, Archbishop of Canterbury, refused confirmation. Representatives of the canons followed Richard I to France, but before they could interview him he died; his successor, King John, received them kindly and granted them permission to hold an election. They were unanimous in their selection of Gerald, and Gerald acted as bishop-elect for much of the next four years; and, as Hubert still refused to confirm the election, Gerald started for Rome to have his election confirmed. There he had an interview with Pope Innocent III. He visited Rome on three occasions (1199–1200; 1201; 1202–3) in support of his claims. But in 1198 the archbishop had forestalled him, and his agents in Rome undermined Gerald's case; and as the pope was not convinced that St Davids was independent of Canterbury, Gerald's mission failed. Gerald had pleaded not only his own cause, but that of St Davids as a Metropolitan archbishopric (and thus of the same status as Canterbury) reviving the earlier claims of Rhygyfarch and Bishop Bernard of St Davids. It was in connexion with this cause that he wrote his books De jure Menevensis Ecclesiâ and De Rebus a Se Gestis. Gerald returned, and his cause was now supported by the Princes of Wales, most notably Llywelyn the Great, and Gruffydd ap Rhys II, while King John, frequently in conflict with the Welsh, warmly espoused the cause of the Archbishop of Canterbury. In 1202, Gerald was accused of stirring up the Welsh to rebellion and was put on trial, but the trial came to nothing as the principal judges were absent. After this long struggle, the chapter of St Davids deserted Gerald, and having been obliged to leave Wales, he fled to Rome. The ports had been closed against him, so he travelled in secret. In April 1203 Pope Innocent III annulled both elections, and Geoffrey of Henlaw was appointed to the See of St Davids, despite the strenuous exertions of Gerald. Travelling back to France, he was briefly imprisoned there for these actions. He was afterwards reconciled with the king and was forced to vow never again to support the primacy of St Davids over Canterbury. The expenses of his unsuccessful election were paid by the crown. Gerald maintained his appointment had been prevented by fear of its possible effect on national politics in Wales. He famously complained in a letter to Innocent III, "Because I am a Welshman am I to be debarred from all preferments in Wales? On the same reasoning so would an Englishman in England, a Frenchman in France, and Italian in Italy. But I am sprung from the Princes of Wales and the Barons of the Marches, and when I see injustice in either race I hate it." At this point he resigned his position as archdeacon of Brecon.

Later life
Gerald spent the remainder of his life in academic study, most probably in Lincoln, producing works of devotional instruction and politics, and revising the works on Ireland and Wales he had written earlier in his life. He spent two years (1204–6) in Ireland with his relatives and made a fourth visit to Rome, purely as a pilgrimage, in 1206. The controversy over St Davids soured his relationship with the crown. In 1216 a baronial plan to put Louis VIII of France on the throne of England in the First Barons' War was warmly welcomed by him. He died in about 1223 in his 77th year, probably in Hereford and he is, according to some accounts, buried at St Davids Cathedral.

There is a statue, by Henry Poole of Gerald in City Hall, Cardiff, and he was included in the vote on 100 Welsh Heroes for his Descriptio Cambriae and Itinerarium Cambriae. His reputation in Ireland, due to his negative portrayal of the Irish, is much less friendly.

Works
Gerald's writings in good-quality Latin, based on a thorough knowledge of Classical authors, reflect experiences gained on his travels as well as his great knowledge of the standard authorities. He was respected as a scholar in his time and afterwards. The noted scholar Edward Augustus Freeman, in his Norman Conquest, said he was "the father of comparative philology," and in the preface to the last volume of Gerald's works in the Rolls Series, he calls him "one of the most learned men of a learned age," "the universal scholar." His writings were prolific, running to about ten volumes in modern printed editions. Gerald was a man of strong opinions whose works are frequently polemical, including bitter attacks on his enemies, but he also had an intense curiosity, recording much valuable detail of everyday life in his ethnographic works.

It is generally agreed today that his most distinguished works are those dealing with Wales and Ireland, with his two books on his beloved Wales the most important: Itinerarium Cambriae and Descriptio Cambriae which tell us much about Welsh history and geography and reflect on the cultural relationship between the Welsh and the English. Gerald, despite his desire for an independent Welsh Church and admiration for parts of Welsh life, was very loyal to Norman Marcher rule, regarding the Normans as more civilised than the Welsh, a feeling reflected in his writings. Professor Davies tells us that Gerald, whom he calls "an admirable story-teller", is the only source for some of the most famous of the Welsh folk tales including the declaration of the old man of Pencader to Henry II which concludes Descriptio Cambriae:

It was Gerald who also wrote (of the Welsh) that "If they would be inseparable, they would be insuperable", and that, unlike the English hirelings, who fight for power or to procure gain or wealth, the Welsh patriots fight for their country. He had pleasant things to say about the poetic talents of his people, too:

Gerald could not have predicted the later perfection of cynghanedd, the complex system of sound correspondence that has characterised the strict-metre poetry of the Welsh for so many centuries and that is still practised today, especially in competitions for the eisteddfod chair. Cynghanedd did not become a formal system with strict rules until the fourteenth century, but its uniquely Welsh forms had been honed for centuries before that.

Finally, in Descriptio Cambriae, Gerald penned the following words that give so much pride to Welsh singers of today, especially those who participate in the immensely popular cymanfaoedd canu (hymn-singing festivals) held throughout Wales and North America:

Another part of the above work, however, is less positive. As Gerald puts it, "an attention to order now requires that, in this second part, we should employ our pen in pointing out those particulars in which it seems to transgress the line of virtue and commendation". David Powel published an abridged version of Itinerarium Cambriae and Descriptio Cambriae in 1585 omitting Gerald's negative comments about the Welsh. Due to translations into English, the first being done by Sir Richard Colt Hoare, Bart., and other translations such as in Everyman's Library and Penguin Classics, Gerald's works on Wales are well known today.

In Gerald's writing on Ireland, his love of music is very evident too.

Gerald's works on Ireland, although invaluable for their detail, are obviously biased, and have been attacked by Irish writers such as Stephen White. The following passage from his Topographia Hibernica shows why the Irish might not always be too enamoured with Gerald's views:

Gerald was a keen and observant student of natural history, but the value of his observations is lessened by credulity and the inability to distinguish fact from legend. He gives a vivid and accurate description of the last colony of the Eurasian beaver in Wales on the Teifi, but spoils it by repeating the legend that beavers castrate themselves to avoid danger. Likewise he gives a good description of an osprey fishing, but adds the mythical detail that the bird has one webbed foot. His description of Irish wildlife has been the subject of much adverse comment for its inaccuracies and lapses into fiction but nonetheless, despite its faults, some have argued that it gives an important glimpse of Irish fauna in the 1180s. Certainly the book has valuable details about Irish birds: while the common kingfisher is now common in Ireland, Gerald states clearly that it was not found there in his time: on the other hand the white-throated dipper, which he had evidently not seen before, was very common in Ireland. He also observed the great numbers of birds of prey in Ireland, including the golden eagle and the Eurasian sparrowhawk, which he said were more numerous in Ireland than in England.

List of works

 * Topographia Hibernica ("Topography of Ireland", 1187)
 * Expugnatio Hibernica ("Conquest of Ireland", 1189)
 * Itinerarium Cambriae ("Journey through Wales", 1191)
 * Descriptio Cambriae ("Description of Wales", 1194)
 * De instructione principis ("Education of a prince")
 * De rebus a se gestis ("Autobiography")
 * De iure et statu Menevensis ecclesiae ("Rights and privileges of the Church of St Davids")
 * Gemma ecclesiastica ("Jewel of the church")
 * Speculum ecclesiae ("Mirror of the church")
 * Symbolum electorum
 * Invectiones
 * Retractationes
 * Speculum duorum ("A mirror of two men") (Described at this link)
 * Life of St Hugh of Lincoln
 * Life of Geoffrey, Archbishop of York
 * Life of St Ethelbert
 * Life of St Remigius
 * Life of St David

Lost works

 * Vita sancti Karadoci ("Life of St Caradoc")
 * De fidei fructu fideique defectu ("About the fruit of faith and [about] the lack of faith")
 * Totius Kambriae mappa ("Map of all Wales", c. 1205)
 * De philosophicis flosculis ("The Flowers of Philosophy")

Online editions

 * The Historical Works of Giraldus Cambrensis, containing The Topography of Ireland, and the History of the Conquest of Ireland, translated by Thomas Forester, and The Itinerary through Wales, and the Description of Wales, translated by Sir Richard Colt Hoare, 1905

In popular culture

 * Gerald's tour of Wales in 1188 was detailed in a 1988 cartoon voiced by comedian Max Boyce.
 * Gerald of Wales was 4th in the series of 8 by Nicholas Crane in Great British Journeys.
 * James Goldman's novel Myself As Witness is written from Gerald of Wales' point of view, though in the novel he is referred to as Giraldus Cambrensis, which means Gerald of Wales in Latin.
 * A train service between Holyhead and Cardiff operated by Transport for Wales Rail is named Gerald of Wales.

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

title: Gerald of Wales


text:
Geoffrey of Monmouth

Geoffrey of Monmouth (Galfridus Monemutensis, Galfridus Arturus; Gruffudd ap Arthur, Sieffre o Fynwy; c. 1095) was a Catholic cleric from Monmouth, Wales, and one of the major figures in the development of British historiography and the popularity of tales of King Arthur. He is best known for his chronicle The History of the Kings of Britain (De gestis Britonum or Historia Regum Britanniae) which was widely popular in its day, being translated into other languages from its original Latin. It was given historical credence well into the 16th century, but is now considered historically unreliable.

Life and career
Geoffrey was born between about 1090 and 1100,   in Wales or the Welsh Marches. He had reached the age of majority by 1129 when he is recorded as witnessing a charter.

Geoffrey refers to himself in his Historia as Galfridus Monemutensis (Geoffrey of Monmouth), which indicates a significant connection to Monmouth, Wales, and may refer to his birthplace. His works attest to some acquaintance with the place-names of the region. Geoffrey was known to his contemporaries as Galfridus Arturus or variants thereof. The "Arthur" in these versions of his name may indicate the name of his father or a nickname based on his scholarly interests.

Earlier scholars assumed that Geoffrey was Welsh or at least spoke Welsh. His knowledge of this language appears to have been slight, however, and there is no evidence that he was of either Welsh or Cambro-Norman descent. He may have come from the same French-speaking elite of the Welsh border country as Gerald of Wales, Walter Map, and Robert, Earl of Gloucester, to whom Geoffrey dedicated versions of his History. Frank Merry Stenton and others have suggested that Geoffrey's parents may have been among the many Bretons who took part in William the Conqueror's conquest and settled in the southeast of Wales. Monmouth had been in the hands of Breton lords since 1075 or 1086, and the names Galfridus and Arthur were more common among the Bretons than the Welsh.

He may have served for a while in the Benedictine Monmouth Priory, but most of his adult life appears to have been spent outside Wales. Between 1129 and 1151, his name appears on six charters in the Oxford area, sometimes styled magister (teacher). He was probably a secular canon of St. George's college. All the charters signed by Geoffrey are also signed by Walter, Archdeacon of Oxford, a canon at that church. Another frequent co-signatory is Ralph of Monmouth, a canon of Lincoln.

Archbishop Theobald of Bec consecrated Geoffrey as Bishop of St Asaph at Lambeth on 24 February 1152, having ordained him a priest at Westminster 10 days before. According to Lewis Thorpe, "There is no evidence that he ever visited his see, and indeed the wars of Owain Gwynedd make this most unlikely." He appears to have died between 25 December 1154 and 24 December 1155 according to Welsh chronicles, when his successor took office.

Works
Geoffrey's structuring and shaping of the Merlin and Arthur myths engendered their vast popularity which continues today, and he is generally viewed by scholars as the major establisher of the Arthurian canon. The History's effect on the legend of King Arthur was so vast that Arthurian works have been categorised as "pre-Galfridian" and "post-Galfridian", depending on whether or not they were influenced by him.

Historia Regum Britanniae
Geoffrey wrote several works in Latin, the language of learning and literature in Europe during the medieval period. His major work was the Historia Regum Britanniae (The History of the Kings of Britain), the work best known to modern readers. It relates the purported history of Britain, from its first settlement by Brutus of Troy, a descendant of Trojan hero Aeneas, to the death of Cadwaladr in the 7th century, covering Julius Caesar's invasions of Britain, Kings Leir and Cymbeline, and one of the earliest developed narratives of King Arthur.

Geoffrey claims in his dedication that the book is a translation of an "ancient book in the British language that told in orderly fashion the deeds of all the kings of Britain", given to him by Walter, Archdeacon of Oxford, but modern historians have dismissed this claim. It is likely, however, that the Archdeacon did furnish Geoffrey with some materials in the Welsh language which helped inspire his work, as Geoffrey's position and acquaintance with him would not have permitted him to fabricate such a claim outright. Much of it is based on the Historia Britonum, a 9th-century Welsh-Latin historical compilation, Bede's Ecclesiastical History of the English People, and Gildas's 6th-century polemic De Excidio et Conquestu Britanniae, expanded with material from bardic oral tradition and genealogical tracts, and embellished by Geoffrey's own imagination. In an exchange of manuscript material for their own histories, Robert of Torigny gave Henry of Huntingdon a copy of History, which both Robert and Henry used uncritically as authentic history and subsequently used in their own works, by which means Geoffrey's fictions became embedded in popular history.

The History of the Kings of Britain is now usually considered a literary forgery containing little reliable history. This has since led many modern scholars to agree with William of Newburgh, who wrote around 1190 that "it is quite clear that everything this man wrote about Arthur and his successors, or indeed about his predecessors from Vortigern onwards, was made up, partly by himself and partly by others."

Other contemporaries were similarly unconvinced by Geoffrey's History. For example, Giraldus Cambrensis recounts the experience of a man possessed by demons: "If the evil spirits oppressed him too much, the Gospel of St John was placed on his bosom, when, like birds, they immediately vanished; but when the book was removed, and the History of the Britons by 'Geoffrey Arthur' [as Geoffrey named himself] was substituted in its place, they instantly reappeared in greater numbers, and remained a longer time than usual on his body and on the book."

Geoffrey's major work was nevertheless widely disseminated throughout medieval Western Europe; Acton Griscom listed 186 extant manuscripts in 1929, and others have been identified since. It enjoyed a significant afterlife in a variety of forms, including translations and adaptations such as Wace's Old Norman-French Roman de Brut, Layamon's Middle English Brut, and several anonymous Middle Welsh versions known as Brut y Brenhinedd ("Brut of the Kings"). where it was generally accepted as a true account.

In 2017, Miles Russell published the initial results of the Lost Voices of Celtic Britain Project established at Bournemouth University. The main conclusion of the study was that the Historia Regum Britanniae appears to contain significant demonstrable archaeological fact, despite being compiled many centuries after the period that it describes. Geoffrey seems to have brought together a disparate mass of source material, including folklore, chronicles, king-lists, dynastic tables, oral tales, and bardic praise poems, some of which was irrevocably garbled or corrupted. In doing so, Geoffrey exercised considerable editorial control, massaging the information and smoothing out apparent inconsistencies in order to create a single grand narrative which fed into the preferred narrative of the Norman rulers of Britain. Much of the information that he used can be shown to be derived from two discrete sources:

 * the orally transmitted, heroic tales of the Catuvellauni and Trinovantes, two essentially pre-Roman tribes inhabiting central south-eastern Britain at the very end of the Iron Age;
 * the king-lists of important post-Roman dynasties that ruled territories in western Britain.

Stretching this source material out, chopping, changing and re-editing it in the process, Geoffrey added not just his own fictions but also additional information culled from Roman and early medieval histories and early medieval writers such as Gildas and Bede.

Other writings
Geoffrey's earliest writing was probably the Prophetiae Merlini (Prophecies of Merlin) which he wrote before 1135, and which appears both independently and incorporated into The History of the Kings of Britain. It consists of a series of obscure prophetic utterances attributed to Merlin which he claimed to have translated from an unspecified language.

The third work attributed to Geoffrey is the hexameter poem Vita Merlini (Life of Merlin), based more closely on traditional material about Merlin than the other works. Here he is known as Merlin of the Woods (Merlinus Sylvestris) or Scottish Merlin (Merlinus Caledonius) and is portrayed as an old man living as a crazed and grief-stricken outcast in the forest. The story is set long after the timeframe of the History's Merlin, but the author tries to synchronise the works with references to the mad prophet's previous dealings with Vortigern and Arthur. The Vita did not circulate widely, and the attribution to Geoffrey appears in only one late 13th-century manuscript, but it contains recognisably Galfridian elements in its construction and content, and most critics recognise it as his.

Editions of the Latin text

 * Hammer, Jacob/ Geoffrey of Monmouth, Historia regum Britanniae, a variant version. Edited by Jacob Hammer. Medieval Academy Books, No. 57 (1951). Medieval Academy Electronic Editions. Geoffrey of Monmouth, Second Variant version of the "Historia Regum Britannie" from Library of Matthew Parker.
 * Historia regum Britanniae, MS CUL Ff.1.25, Cambridge Digital Library.
 * Lewis E 247 Gesta regum Anglorum (Deeds of the English Kings); Historia regum Britanniae (History of the Kings of Britain) at OPenn
 * Lewis E 247 Gesta regum Anglorum (Deeds of the English Kings); Historia regum Britanniae (History of the Kings of Britain) at OPenn

English translations available on the internet

 * Historia Regum Britanniae:
 * Histories of the Kings of Britain, tr. by Sebastian Evans, at Sacred Texts
 * By Aaron Thompson with revisions by J. A. Giles at http://www.yorku.ca/inpar/geoffrey_thompson.pdf. (PDF)
 * (Arthurian passages only) edited and translated by J. A. Giles at http://www.lib.rochester.edu/camelot/geofhkb.htm.
 * Vita Merlini, Basil Clarke's English translation from Life of Merlin: Vita Merlini (Cardiff: University of Wales Press, 1973).
 * At Jones the Celtic Encyclopedia
 * At Sacred-texts.com

title: Geoffrey of Monmouth


text:
Llandaff

Llandaff (Llandaf ; from llan 'church' and Taf) is a district, community and coterminous electoral ward in the north of Cardiff, capital of Wales. It was incorporated into the city in 1922. It is the seat of the Bishop of Llandaff, whose diocese within the Church in Wales covers the most populous area of Wales.

History

Most of the history of Llandaff centres on its role as a religious site. Before the creation of Llandaff Cathedral, it became established as a Christian place of worship in the 6th century AD, probably because of its location as the first firm ground north of the point where the river Taff met the Bristol Channel, and because of its pre-Christian location as a river crossing on a north–south trade route. Evidence of Romano-British ritual burials have been found under the present cathedral. The date of the moving of the cathedral to Llandaff is disputed, but elements of the fabric date from the 12th century, such as the impressive Romanesque Urban Arch, named after the 12th century Bishop, Urban. It has had a history of continual destruction and restoration, as a result of warfare, neglect, and natural disaster. Llandaff has been a focal point of devastating attacks by Owain Glyndŵr and Oliver Cromwell. It was the second most damaged cathedral in the UK (after Coventry Cathedral), following Luftwaffe bombing during World War II, and subsequently restored by the architect George Pace. One of its main modern points of interest is the aluminium figure of Christ in Majesty (1954–5), by Jacob Epstein, which is suspended above the nave. In 2007, a lightning strike to its spire sent a surge through the building; which destroyed its organ. Its replacement, the largest to be built in the UK for over 40 years, was inaugurated in 2010.

The Bishop's Palace, also known as Llandaff Castle or Bishop's Castle now in ruins, lies to the south of the cathedral. It is believed it was constructed at a similar date to Caerphilly Castle, in the late 13th century. It is also believed it was abandoned after being attacked and damaged by Glyndŵr in the 15th century. The gatehouse of the Palace survives, and the courtyard is now a public garden.

Llandaff never developed into a chartered borough, and by the 19th century, was described as "reduced to a mere village... It consists of little more than two short streets of cottages, not lighted or paved, terminating in a square, into which the great gateway of the old palace formerly opened, and where are still several genteel houses."

Historically, Llandaff was informally known as a 'city', because of its status as the seat of the Bishop of Llandaff. This city status was never officially recognised, largely because the community did not possess a charter of incorporation. The ancient parish of Llandaff included a wide area. Apart from Llandaff itself, it included the townships of Canton, Ely, Fairwater, and Gabalfa.

During the development of the South Wales coalfield and Cardiff Docks, the parish was gradually absorbed into the Borough of Cardiff during the 19th and 20th centuries. Seen as a clean and green up-market countrified village location close to the fast developing city, many of the better-off coal merchants and business people chose to live in Llandaff, including the Insole family. The house now known as Insole Court dates originally from 1856. Llandaff itself became a civil parish, and from 1894 to 1922, was part of the Llandaff and Dinas Powis Rural District. On 9 November 1922, the county borough of Cardiff was extended to include the area.

Demography
At the United Kingdom Census 2011, the population of the Llandaff was 8,997; of whom 4,309 were male, and 4,688 female. 91.6% were recorded as being of various white ethnicities. Approximately 65% of the population were returned as Christian, with about 1.5% each being Hindu or Muslim, and 30% having no religion or no stated religion.

Welsh language
In the 2011 census, 15.3% of the population over 3 years old in Llandaff were recorded as speaking Welsh, or 1,337 people. This was a small drop compared to the 2001 census figure, which was 15.4%.

Broadcasting House in Llandaff was the headquarters of BBC Cymru Wales until it relocated to Central Square in Cardiff between October 2019 and July 2020.

Research by Owen John Thomas shows the historical strength of the Welsh language in Llandaff. According to his book: Yr Iaith Gymraeg yng Nghaerdydd c. 1800–1914 (The Welsh language in Cardiff c. 1800–1914), the nonconformist church in Cardiff Road was a Welsh-language church in 1813. His work also shows that Welsh was the main language of the street in Llandaff in the 17th century.

Governance
Llandaff is both an electoral ward, and a community (cymuned) of the City of Cardiff. There is no community council for the area. The electoral ward of Llandaff is bounded by Radyr & Morganstown to the north west; Llandaff North to the north; Riverside to the south east; Canton to the south; and Fairwater to the west. The ward is represented by two councillors on Cardiff Council, Sean Driscoll and Peter Huw Jenkins, the former is a member of the Conservative Party, the latter is a member of the Labour Party.

In the UK Parliament, Llandaff is part of the constituency of Cardiff West. Its most prominent MPs were former Speaker of the House of Commons; George Thomas, and former First Minister of Wales and Welsh Labour Party leader Rhodri Morgan. The current MP is Labour's Kevin Brennan, elected in 2001. In the Senedd, Llandaff is part of the constituency of Cardiff West, whose MS since 2011 is Mark Drakeford of Labour; he succeeded Rhodri Morgan upon the latter's retirement. The constituency is within the electoral region of South Wales Central, whose four current MSs are Conservatives Andrew RT Davies and Joel James; Plaid Cymru's Rhys ab Owen and Heledd Fychan.

Higher education

 * Cardiff Metropolitan University, Llandaff campus
 * St. Michael's College, Anglican theological college

State secondary schools

 * Bishop of Llandaff Church in Wales High School, comprehensive, English medium.
 * Ysgol Gyfun Gymraeg Glantaf (serving Llandaff, but located in Llandaff North), Welsh medium.

State primary schools

 * Danescourt Primary School, English medium.
 * Llandaff City Church in Wales Primary School, English medium.
 * Ysgol Pencae, Welsh medium.

Independent schools

 * The Cathedral School, ages 3–18 co-educational, English medium.
 * Howell's School, ages 3–18 girls, with co-educational sixth form, English medium.

Transport
Llandaff is served by railway stations at Danescourt, Fairwater, and Waun-Gron Park; each is about a mile from the cathedral. There is a half-hourly service to and from Cardiff on the Cardiff City Line. Llandaf railway station is located in Llandaff North.

Cardiff Bus services 1/2 (City Circle), 24/25 (Whitchurch), 62/63 (Radyr/Morganstown), 64/65 (Heath Hospital/Llanrumney), 66 (Danescourt), and Stagecoach service 122 (Tonypandy) operate through the area to/from Cardiff city centre.

Western Avenue (A48) runs through the south of the area, heading eastbound to Gabalfa and M4 J29, and westbound to Ely, Culverhouse Cross, and M4 J33. Cardiff Road leads south towards Cardiff city centre.

Economy
The major employment sectors in the area are:
 * Education (16.2%)
 * Human health and social work activities (13.8%)
 * Wholesale and retail trade; repair of motor vehicles and motor cycles (11.3%)
 * Public administration and defence (9.6%)
 * Professional, scientific and technical activities (8.9%)
 * Other services (6.2%)
 * Construction (6.1%)
 * Information and communication (5.6%)
 * Banking, finance and insurance (5.3%)
 * Accommodation and food service activities (4.3%)
 * Manufacturing (4%)

Broadcasting House, formerly the headquarters of BBC Cymru Wales, was opened in Llandaff in 1966. BBC Cymru Wales moved to new facilities at Central Square, Cardiff in 2020, with the remaining Llandaff site due for housing development.

Doctor Who
Following its revival in 2005, the long-running science-fiction television series Doctor Who was produced by BBC Wales in Llandaff. Production was relocated to the BBC's new Roath Lock studios in Cardiff Bay in 2012. The location scenes of four episodes were filmed in Llandaff:
 * "Human Nature" (2007)
 * "The Family of Blood" (2007)
 * "The Eleventh Hour" (2010)
 * "Vincent and the Doctor" (2010)

Notable people

 * Sir Ivor Atkins (1869–1953); choirmaster and organist, was born in Llandaff.
 * James Dean Bradfield (born 1969), singer, guitarist and songwriter with the Manic Street Preachers
 * Charlotte Church (born 1986), singer and television presenter; was born in Llandaff, and attended Howell's School.
 * Roald Dahl (1916–1990), author; was born in Llandaff, and attended the Cathedral School.
 * Cheryl Gillan (1952–2021), Secretary of State for Wales (2010–2012); was born in Llandaff.
 * Francis Lewis (1713–1802); one of the signatories of the United States Declaration of Independence.
 * Sir David Mathew (Dafydd ap Mathew) (1400–1484), Standard Bearer of England to Edward IV; fought in the Battle of Towton, and is credited for saving King Edward IV's life, and granted the right to use 'Towton' on the Mathew Family arms. His tomb is in Llandaff Cathedral.
 * Terry Nation (1930−1997), television scriptwriter was born in Llandaff, wrote seventy Doctor Who episodes, and created the Daleks. He also created the BBC series Blake's 7 and Survivors and wrote a number of episodes for Lew Grade's ITC Entertainment show The Saint.
 * Ivor Williams (1908–1982), artist; lived and painted in Llandaff.

title: Llandaff


text:
Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.

Definition
If V is a vector space over a field K and if W is a subset of V, then W is a linear subspace of V if under the operations of V, W is a vector space over K. Equivalently, a nonempty subset W is a linear subspace of V if, whenever $w_{1}, w_{2}$ are elements of W and $α, β$ are elements of K, it follows that $αw_{1} + βw_{2}$ is in W.

As a corollary, all vector spaces are equipped with at least two (possibly different) linear subspaces: the zero vector space consisting of the zero vector alone and the entire vector space itself. These are called the trivial subspaces of the vector space.

Example I
In the vector space V = R3 (the real coordinate space over the field R of real numbers), take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V.

Proof:
 * 1) Given u and v in W, then they can be expressed as u = (u1, u2, 0) and v = (v1, v2, 0). Then u + v = (u1+v1, u2+v2, 0+0) = (u1+v1, u2+v2, 0). Thus, u + v is an element of W, too.
 * 2) Given u in W and a scalar c in R, if u = (u1, u2, 0) again, then cu = (cu1, cu2, c0) = (cu1, cu2,0). Thus, cu is an element of W too.

Example II
Let the field be R again, but now let the vector space V be the Cartesian plane R2. Take W to be the set of points (x, y) of R2 such that x = y. Then W is a subspace of R2.

Proof:
 * 1) Let p = (p1, p2) and q = (q1, q2) be elements of W, that is, points in the plane such that p1 = p2 and q1 = q2. Then p + q = (p1+q1, p2+q2); since p1 = p2 and q1 = q2, then p1 + q1 = p2 + q2, so p + q is an element of W.
 * 2) Let p = (p1, p2) be an element of W, that is, a point in the plane such that p1 = p2, and let c be a scalar in R. Then cp = (cp1, cp2); since p1 = p2, then cp1 = cp2, so cp is an element of W.

In general, any subset of the real coordinate space Rn that is defined by a homogeneous system of linear equations will yield a subspace. (The equation in example I was z = 0, and the equation in example II was x = y.)

Example III
Again take the field to be R, but now let the vector space V be the set RR of all functions from R to R. Let C(R) be the subset consisting of continuous functions. Then C(R) is a subspace of RR.

Proof:
 * 1) We know from calculus that 0 ∈ C(R) ⊂ RR.
 * 2) We know from calculus that the sum of continuous functions is continuous.
 * 3) Again, we know from calculus that the product of a continuous function and a number is continuous.

Example IV
Keep the same field and vector space as before, but now consider the set Diff(R) of all differentiable functions. The same sort of argument as before shows that this is a subspace too.

Examples that extend these themes are common in functional analysis.

Properties of subspaces
From the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples. Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set W is a subspace if and only if every linear combination of finitely many elements of W also belongs to W. The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time.

In a topological vector space X, a subspace W need not be topologically closed, but a finite-dimensional subspace is always closed. The same is true for subspaces of finite codimension (i.e., subspaces determined by a finite number of continuous linear functionals).

Descriptions
Descriptions of subspaces include the solution set to a homogeneous system of linear equations, the subset of Euclidean space described by a system of homogeneous linear parametric equations, the span of a collection of vectors, and the null space, column space, and row space of a matrix. Geometrically (especially over the field of real numbers and its subfields), a subspace is a flat in an n-space that passes through the origin.

A natural description of a 1-subspace is the scalar multiplication of one non-zero vector v to all possible scalar values. 1-subspaces specified by two vectors are equal if and only if one vector can be obtained from another with scalar multiplication:
 * $$\exist c\in K: \mathbf{v}' = c\mathbf{v}\text{ (or }\mathbf{v} = \frac{1}{c}\mathbf{v}'\text{)}$$

This idea is generalized for higher dimensions with linear span, but criteria for equality of k-spaces specified by sets of k vectors are not so simple.

A dual description is provided with linear functionals (usually implemented as linear equations). One non-zero linear functional F specifies its kernel subspace F = 0 of codimension 1. Subspaces of codimension 1 specified by two linear functionals are equal, if and only if one functional can be obtained from another with scalar multiplication (in the dual space):
 * $$\exist c\in K: \mathbf{F}' = c\mathbf{F}\text{ (or }\mathbf{F} = \frac{1}{c}\mathbf{F}'\text{)}$$

It is generalized for higher codimensions with a system of equations. The following two subsections will present this latter description in details, and the remaining four subsections further describe the idea of linear span.

Systems of linear equations
The solution set to any homogeneous system of linear equations with n variables is a subspace in the coordinate space Kn: $$\left\{ \left[\!\! \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \!\!\right] \in K^n : \begin{alignat}{6} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = 0&   \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = 0&   \\ &&    &&            &&              &&                && \vdots\quad& \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = 0& \end{alignat} \right\}. $$

For example, the set of all vectors $(x, y, z)$ (over real or rational numbers) satisfying the equations $$x + 3y + 2z = 0 \quad\text{and}\quad 2x - 4y + 5z = 0$$ is a one-dimensional subspace. More generally, that is to say that given a set of n independent functions, the dimension of the subspace in Kk will be the dimension of the null set of A, the composite matrix of the n functions.

Null space of a matrix
In a finite-dimensional space, a homogeneous system of linear equations can be written as a single matrix equation:

 * $$A\mathbf{x} = \mathbf{0}.$$

The set of solutions to this equation is known as the null space of the matrix. For example, the subspace described above is the null space of the matrix

 * $$A = \begin{bmatrix} 1 & 3 & 2 \\ 2 & -4 & 5 \end{bmatrix} .$$

Every subspace of Kn can be described as the null space of some matrix (see below for more).

Linear parametric equations
The subset of Kn described by a system of homogeneous linear parametric equations is a subspace:

 * $$\left\{ \left[\!\! \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \!\!\right] \in K^n : \begin{alignat}{7}

x_1 &&\; = \;&& a_{11} t_1 &&\; + \;&& a_{12} t_2 &&\; + \cdots + \;&& a_{1m} t_m &   \\ x_2 &&\; = \;&& a_{21} t_1 &&\; + \;&& a_{22} t_2 &&\; + \cdots + \;&& a_{2m} t_m &   \\ && \vdots\;\; &&      &&       &&            &&                &&            &    \\ x_n &&\; = \;&& a_{n1} t_1 &&\; + \;&& a_{n2} t_2 &&\; + \cdots + \;&& a_{nm} t_m &   \\ \end{alignat} \text{ for some } t_1,\ldots,t_m\in K \right\}. $$

For example, the set of all vectors (x, y, z) parameterized by the equations

 * $$x = 2t_1 + 3t_2,\;\;\;\;y = 5t_1 - 4t_2,\;\;\;\;\text{and}\;\;\;\;z = -t_1 + 2t_2$$

is a two-dimensional subspace of K3, if K is a number field (such as real or rational numbers).

Span of vectors
In linear algebra, the system of parametric equations can be written as a single vector equation:

 * $$\begin{bmatrix} x \\ y \\ z \end{bmatrix} \;=\; t_1 \!\begin{bmatrix} 2 \\ 5 \\ -1 \end{bmatrix} + t_2 \!\begin{bmatrix} 3 \\ -4 \\ 2 \end{bmatrix}.$$

The expression on the right is called a linear combination of the vectors (2, 5, −1) and (3, −4, 2). These two vectors are said to span the resulting subspace.

In general, a linear combination of vectors v1, v2, ... , vk is any vector of the form

 * $$t_1 \mathbf{v}_1 + \cdots + t_k \mathbf{v}_k.$$

The set of all possible linear combinations is called the span:

 * $$\text{Span} \{ \mathbf{v}_1, \ldots, \mathbf{v}_k \}

= \left\{ t_1 \mathbf{v}_1 + \cdots + t_k \mathbf{v}_k : t_1,\ldots,t_k\in K \right\} .$$

If the vectors v1, ..., vk have n components, then their span is a subspace of Kn. Geometrically, the span is the flat through the origin in n-dimensional space determined by the points v1, ... , vk.

 * Example
 * The xz-plane in R3 can be parameterized by the equations
 * $$x = t_1, \;\;\; y = 0, \;\;\; z = t_2.$$

 * As a subspace, the xz-plane is spanned by the vectors (1, 0, 0) and (0, 0, 1). Every vector in the xz-plane can be written as a linear combination of these two:

 * $$(t_1, 0, t_2) = t_1(1,0,0) + t_2(0,0,1)\text{.}$$

 * Geometrically, this corresponds to the fact that every point on the xz-plane can be reached from the origin by first moving some distance in the direction of (1, 0, 0) and then moving some distance in the direction of (0, 0, 1).

Column space and row space
A system of linear parametric equations in a finite-dimensional space can also be written as a single matrix equation:

 * $$\mathbf{x} = A\mathbf{t}\;\;\;\;\text{where}\;\;\;\;A = \left[ \begin{alignat}{2} 2 && 3 & \\ 5 && \;\;-4 & \\ -1 && 2 & \end{alignat} \,\right]\text{.}$$

In this case, the subspace consists of all possible values of the vector x. In linear algebra, this subspace is known as the column space (or image) of the matrix A. It is precisely the subspace of Kn spanned by the column vectors of A.

The row space of a matrix is the subspace spanned by its row vectors. The row space is interesting because it is the orthogonal complement of the null space (see below).

Independence, basis, and dimension
In general, a subspace of Kn determined by k parameters (or spanned by k vectors) has dimension k. However, there are exceptions to this rule. For example, the subspace of K3 spanned by the three vectors (1, 0, 0), (0, 0, 1), and (2, 0, 3) is just the xz-plane, with each point on the plane described by infinitely many different values of t1, t2, t3.

In general, vectors v1, ... , vk are called linearly independent if

 * $$t_1 \mathbf{v}_1 + \cdots + t_k \mathbf{v}_k \;\ne\; u_1 \mathbf{v}_1 + \cdots + u_k \mathbf{v}_k$$

for (t1, t2, ..., tk) ≠ (u1, u2, ... , uk). If v1, ..., vk are linearly independent, then the coordinates t1, ..., tk for a vector in the span are uniquely determined.

A basis for a subspace S is a set of linearly independent vectors whose span is S. The number of elements in a basis is always equal to the geometric dimension of the subspace. Any spanning set for a subspace can be changed into a basis by removing redundant vectors (see § Algorithms below for more).

 * Example
 * Let S be the subspace of R4 defined by the equations
 * $$x_1 = 2 x_2\;\;\;\;\text{and}\;\;\;\;x_3 = 5x_4.$$
 * Then the vectors (2, 1, 0, 0) and (0, 0, 5, 1) are a basis for S. In particular, every vector that satisfies the above equations can be written uniquely as a linear combination of the two basis vectors:

 * $$(2t_1, t_1, 5t_2, t_2) = t_1(2, 1, 0, 0) + t_2(0, 0, 5, 1).$$

 * The subspace S is two-dimensional. Geometrically, it is the plane in R4 passing through the points (0, 0, 0, 0), (2, 1, 0, 0), and (0, 0, 5, 1).

Inclusion
The set-theoretical inclusion binary relation specifies a partial order on the set of all subspaces (of any dimension).

A subspace cannot lie in any subspace of lesser dimension. If dim U = k, a finite number, and U ⊂ W, then dim W = k if and only if U = W.

Intersection
Given subspaces U and W of a vector space V, then their intersection U ∩ W := {v ∈ V : v is an element of both U and W} is also a subspace of V.

Proof:
 * 1) Let v and w be elements of U ∩ W. Then v and w belong to both U and W. Because U is a subspace, then v + w belongs to U. Similarly, since W is a subspace, then v + w belongs to W. Thus, v + w belongs to U ∩ W.
 * 2) Let v belong to U ∩ W, and let c be a scalar. Then v belongs to both U and W. Since U and W are subspaces, cv belongs to both U and W.
 * 3) Since U and W are vector spaces, then 0 belongs to both sets. Thus, 0 belongs to U ∩ W.

For every vector space V, the set {0} and V itself are subspaces of V.

Sum
If U and W are subspaces, their sum is the subspace $$U + W = \left\{ \mathbf{u} + \mathbf{w} \colon \mathbf{u}\in U, \mathbf{w}\in W \right\}.$$

For example, the sum of two lines is the plane that contains them both. The dimension of the sum satisfies the inequality $$\max(\dim U,\dim W) \leq \dim(U + W) \leq \dim(U) + \dim(W).$$

Here, the minimum only occurs if one subspace is contained in the other, while the maximum is the most general case. The dimension of the intersection and the sum are related by the following equation: $$\dim(U+W) = \dim(U) + \dim(W) - \dim(U \cap W).$$

A set of subspaces is independent when the only intersection between any pair of subspaces is the trivial subspace. The direct sum is the sum of independent subspaces, written as $$U \oplus W$$. An equivalent restatement is that a direct sum is a subspace sum under the condition that every subspace contributes to the span of the sum.

The dimension of a direct sum $$U \oplus W$$ is the same as the sum of subspaces, but may be shortened because the dimension of the trivial subspace is zero.

$$\dim (U \oplus W) = \dim (U) + \dim (W)$$

Lattice of subspaces
The operations intersection and sum make the set of all subspaces a bounded modular lattice, where the {0} subspace, the least element, is an identity element of the sum operation, and the identical subspace V, the greatest element, is an identity element of the intersection operation.

Orthogonal complements
If $$V$$ is an inner product space and $$N$$ is a subset of $$V$$, then the orthogonal complement of $$N$$, denoted $$N^{\perp}$$, is again a subspace. If $$V$$ is finite-dimensional and $$N$$ is a subspace, then the dimensions of $$N$$ and $$N^{\perp}$$ satisfy the complementary relationship $$\dim (N) + \dim (N^{\perp}) = \dim (V) $$. Moreover, no vector is orthogonal to itself, so $$ N \cap N^\perp = \{ 0 \}$$ and $$V$$ is the direct sum of $$N$$ and $$N^{\perp}$$. Applying orthogonal complements twice returns the original subspace: $$(N^{\perp})^{\perp} = N$$ for every subspace $$N$$.

This operation, understood as negation ($$\neg$$), makes the lattice of subspaces a (possibly infinite) orthocomplemented lattice (although not a distributive lattice).

In spaces with other bilinear forms, some but not all of these results still hold. In pseudo-Euclidean spaces and symplectic vector spaces, for example, orthogonal complements exist. However, these spaces may have null vectors that are orthogonal to themselves, and consequently there exist subspaces $$N$$ such that $$N \cap N^{\perp} \ne \{ 0 \}$$. As a result, this operation does not turn the lattice of subspaces into a Boolean algebra (nor a Heyting algebra).

Algorithms
Most algorithms for dealing with subspaces involve row reduction. This is the process of applying elementary row operations to a matrix, until it reaches either row echelon form or reduced row echelon form. Row reduction has the following important properties:
 * 1) The reduced matrix has the same null space as the original.
 * 2) Row reduction does not change the span of the row vectors, i.e. the reduced matrix has the same row space as the original.
 * 3) Row reduction does not affect the linear dependence of the column vectors.

Basis for a row space

 * Input An m × n matrix A.
 * Output A basis for the row space of A.
 * Use elementary row operations to put A into row echelon form.
 * The nonzero rows of the echelon form are a basis for the row space of A.

See the article on row space for an example.

If we instead put the matrix A into reduced row echelon form, then the resulting basis for the row space is uniquely determined. This provides an algorithm for checking whether two row spaces are equal and, by extension, whether two subspaces of Kn are equal.

Subspace membership

 * Input A basis {b1, b2, ..., bk} for a subspace S of Kn, and a vector v with n components.
 * Output Determines whether v is an element of S
 * Create a (k + 1) × n matrix A whose rows are the vectors b1, ..., bk and v.
 * Use elementary row operations to put A into row echelon form.
 * If the echelon form has a row of zeroes, then the vectors {b1, ..., bk, v} are linearly dependent, and therefore v ∈ S.

Basis for a column space

 * Input An m × n matrix A
 * Output A basis for the column space of A
 * Use elementary row operations to put A into row echelon form.
 * Determine which columns of the echelon form have pivots. The corresponding columns of the original matrix are a basis for the column space.

See the article on column space for an example.

This produces a basis for the column space that is a subset of the original column vectors. It works because the columns with pivots are a basis for the column space of the echelon form, and row reduction does not change the linear dependence relationships between the columns.

Coordinates for a vector

 * Input A basis {b1, b2, ..., bk} for a subspace S of Kn, and a vector v ∈ S
 * Output Numbers t1, t2, ..., tk such that v = t1b1 + ··· + tkbk
 * Create an augmented matrix A whose columns are b1,...,bk, with the last column being v.
 * Use elementary row operations to put A into reduced row echelon form.
 * Express the final column of the reduced echelon form as a linear combination of the first k columns. The coefficients used are the desired numbers t1, t2, ..., tk. (These should be precisely the first k entries in the final column of the reduced echelon form.)

If the final column of the reduced row echelon form contains a pivot, then the input vector v does not lie in S.

Basis for a null space

 * Input An m × n matrix A.
 * Output A basis for the null space of A
 * Use elementary row operations to put A in reduced row echelon form.
 * Using the reduced row echelon form, determine which of the variables x1, x2, ..., xn are free. Write equations for the dependent variables in terms of the free variables.
 * For each free variable xi, choose a vector in the null space for which xi = 1 and the remaining free variables are zero. The resulting collection of vectors is a basis for the null space of A.

See the article on null space for an example.

Basis for the sum and intersection of two subspaces
Given two subspaces $U$ and $W$ of $V$, a basis of the sum $$U + W$$ and the intersection $$U \cap W$$ can be calculated using the Zassenhaus algorithm.

Equations for a subspace

 * Input A basis {b1, b2, ..., bk} for a subspace S of Kn
 * Output An (n − k) × n matrix whose null space is S.
 * Create a matrix A whose rows are b1, b2, ..., bk.
 * Use elementary row operations to put A into reduced row echelon form.
 * Let c1, c2, ..., cn be the columns of the reduced row echelon form. For each column without a pivot, write an equation expressing the column as a linear combination of the columns with pivots.
 * This results in a homogeneous system of n − k linear equations involving the variables c1,...,cn. The (n − k) × n matrix corresponding to this system is the desired matrix with nullspace S.

 * Example
 * If the reduced row echelon form of A is

 * $$\left[ \begin{alignat}{6}

1 && 0 && -3 && 0 && 2 && 0 \\ 0 && 1 &&  5 && 0 && -1 && 4 \\ 0 && 0 &&  0 && 1 &&  7 && -9 \\ 0 && \;\;\;\;\;0 &&  \;\;\;\;\;0 && \;\;\;\;\;0 &&  \;\;\;\;\;0 && \;\;\;\;\;0 \end{alignat} \,\right] $$

 * then the column vectors c1, ..., c6 satisfy the equations

 * $$ \begin{alignat}{1}

\mathbf{c}_3 &= -3\mathbf{c}_1 + 5\mathbf{c}_2 \\ \mathbf{c}_5 &= 2\mathbf{c}_1 - \mathbf{c}_2 + 7\mathbf{c}_4 \\ \mathbf{c}_6 &= 4\mathbf{c}_2 - 9\mathbf{c}_4 \end{alignat}$$

 * It follows that the row vectors of A satisfy the equations

 * $$ \begin{alignat}{1}

x_3 &= -3x_1 + 5x_2 \\ x_5 &= 2x_1 - x_2 + 7x_4 \\ x_6 &= 4x_2 - 9x_4. \end{alignat}$$

 * In particular, the row vectors of A are a basis for the null space of the corresponding matrix.

title: Linear subspace


text:
Mangifera

Mangifera is a genus of flowering plants in the cashew family, Anacardiaceae. It contains about 69 species, with the best-known being the common mango (Mangifera indica). The center of diversity of the genus is in the Malesian ecoregion of Southeast Asia, particularly in Sumatra, Borneo, and the Malay Peninsula. They are generally canopy trees in lowland rainforests, reaching a height of 30 - 40 m.

Uses
Mangifera species are widely cultivated in Asia and elsewhere. More than 27 species in the genus bear edible, fleshy fruits, especially the common mango (M. indica). Others, such as M. foetida, yield astringent fruits that can be eaten pickled.

Mango wastes, such as the seed kernel and peel, have high functional and nutritional potential. Mango seed contains important bioactive compounds that have high antioxidant activity, lipids that have acceptable physical and chemical characteristics (free of trans fatty acids), and a somewhat high (about 6%) protein content. The mango peel contains considerable amounts of antioxidants and dietary fiber.

Fossil record
The earliest fossil species thought to be related to Mangifera is Eomangiferophyllum damalgiriense from the upper Paleocene of northeastern India. Leaves have also been reported from the Paleocene of Japan and the Eocene of Germany, though these have been considered questionable. Leaf fossils confidently assignable to Mangifera have been collected from Oligocene or early Miocene sediments in northern Thailand.

Subgenera

 * Subgenus Mangifera
 * Section Marchandora Pierre
 * Section Euantherae Pierre
 * Section Rawa Kosterm.
 * Section Mangifera Kosterm.
 * Subgenus Limus (Marchand) Kosterm.

Species
These species are accepted:

 * Mangifera acutigemma Kosterm.
 * Mangifera altissima Blanco
 * Mangifera andamanica King
 * ''Mangifera applanata Kosterm.
 * Mangifera austroindica Kosterm.
 * Mangifera blommesteinii Kosterm.
 * Mangifera bullata Kosterm.
 * Mangifera caesia Jack
 * Mangifera campnospermoides Kosterm.
 * Mangifera camptosperma Pierre
 * Mangifera casturi Kosterm.
 * Mangifera cochinchinensis Engl. (Indo-China)
 * Mangifera collina Kosterm.
 * Mangifera decandra Ding Hou
 * Mangifera dewildei Kosterm.
 * Mangifera dongnaiensis Pierre
 * Mangifera duperreana Pierre
 * Mangifera flava Evrard
 * Mangifera foetida Lour.
 * Mangifera gedebi Miq.
 * Mangifera gracilipes Hook.f.
 * Mangifera griffithii Hook.f.
 * Mangifera indica L. : the common mango (type sp.) (synonym M. austroyunnanensis Hu )
 * Mangifera inocarpoides Merr. & L.M.Perry
 * Mangifera khasiana Pierre
 * Mangifera khoonmengiana Kochummen
 * Mangifera lagenifera Griff.
 * Mangifera lalijiwa Kosterm.
 * Mangifera laurina Blume
 * Mangifera linearifolia (Mukh.) Kosterm.
 * Mangifera macrocarpa Blume
 * Mangifera magnifica Kochummen
 * Mangifera mariana Buch.-Ham.
 * Mangifera merrillii Mukherji
 * Mangifera minor Blume
 * Mangifera minutifolia Evrard
 * Mangifera monandra Merr.
 * Mangifera nicobarica Kosterm.
 * Mangifera odorata Griff.
 * Mangifera orophila Kosterm.
 * Mangifera pajang Kosterm.
 * Mangifera paludosa Kosterm.
 * Mangifera parvifolia Boerl. & Koord.
 * Mangifera pedicellata Kosterm.
 * Mangifera pentandra Hook.f.
 * Mangifera persiciforma C.Y. Wu & T.L. Ming : (synonyms: M. persiciformis C.Y. Wu & T.L. Ming, Mangifera hiemalis J.Y. Liang )
 * Mangifera pseudoindica Kosterm.
 * Mangifera quadrifida Jack ex Wall.
 * Mangifera reba Pierre.
 * Mangifera rubropetala Kosterm.
 * Mangifera rufocostata Kosterm.
 * Mangifera similis Blume (synonym: Mangifera torquenda Kosterm. )
 * Mangifera subsessilifolia Kosterm.
 * Mangifera sulavesiana Kosterm.
 * Mangifera sumatrana Miq.
 * Mangifera sumbawaensis Kosterm.
 * Mangifera superba Hook.f.
 * Mangifera swintonioides Kosterm.
 * Mangifera sylvatica Roxb.
 * Mangifera timorensis Blume
 * Mangifera transversalis Kosterm.
 * Mangifera zeylanica (Blume) Hook.f.

Formerly placed here

 * Bouea oppositifolia (Roxb.) Meisn. (as M. oppositifolia Roxb. )
 * Buchanania cochinchinensis (Lour.) M.R.Almeida (as M. axillaris Desr. )
 * Elaeodendron glaucum (Rottb.) Pers. (as M. glauca Rottb. )
 * Irvingia gabonensis (Aubry-Lecomte ex O'Rorke) Baill. (as M. gabonensis Aubry-Lecomte ex O'Rorke )
 * Fegimanra africana (Oliv.) Pierre. (as M. africana Oliv. )
 * Spondias pinnata (J.Koenig ex L.f.) Kurz (as M. pinnata J.Koenig ex L.f. )

title: Mangifera