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Talk:Arthur (TV series)

"Suggested Changes to the Critical Response Section"
I suggest a couple of changes to the "Critical Response" section under "Reception".

1. Make it explicitly clear that Ostrov, Gentile and Crick's original study suggests that all T.V. (including "educational" shows) can lead to increased "relational aggression". They only used Arthur as an example of an educational show and certainly don't make a causative statement that Arthur leads to relational aggression in children. The reason for why educational media might lead to relational aggression is also only a hypothesis (not tested in the study) and only applies to young children (4-6 years) who are unable to understand the lesson the show is trying to impart and instead pick up on the behaviours the show is trying to discourage.

2. Completely remove the quote from NurtureShock ("Essentially, Ostrov had just found that Arthur is more dangerous for children than Power Rangers"). I has a brief look at the book's authors (neither of them have a background in child psychology or any scientific field - both of them appear to be pop-science writers) and the Amazon page and it looks like a Gladwell-ian book concerned with creating simple straightforward narrative for popular consumption. This explains the mischaracterization "Essentially, Ostrov had just found that Arthur is more dangerous for children than Power Rangers" which should not be someone's takeaway if you actually read the journal article. The book also states that "educational" media exposure was found to increase physical aggression in a follow up study but the only follow up study I found summarizes that "Results replicate and extend prior research that has demonstrated links between educational media exposure and relational aggression, but not physical aggression, during early childhood." (emphasis mine).

Link to the journal article: http://ubwp.buffalo.edu/socialdevlab/wp-content/uploads/sites/32/2020/06/Ostrov-Gentile-Crick-2006-Social-Development.pdf

Follow up study from 2012: https://ubwp.buffalo.edu/socialdevlab/wp-content/uploads/sites/32/2015/09/Evaluating-the-effect-of-educational-media-exposure-on-aggression-in-early-childhood..pdf TranquilDragon (talk) 09:17, 20 December 2022 (UTC)

"Arthur the tv show" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Arthur_the_tv_show&redirect=no Arthur the tv show] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. Sincerely, --AugustusAudax (talk|contribs) P.S: Aliens exist 19:56, 30 May 2023 (UTC)

"Clark Caves" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Clark_Caves&redirect=no Clark Caves] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. –CopperyMarrow15 (talk | edits) Feel free to ping me! 03:06, 14 October 2023 (UTC)

"Kresblain" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Kresblain&redirect=no Kresblain] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. –CopperyMarrow15 (talk | edits) Feel free to ping me! 03:06, 14 October 2023 (UTC)

"Kresblain the Merry Magician" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=Kresblain_the_Merry_Magician&redirect=no Kresblain the Merry Magician] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. –CopperyMarrow15 (talk | edits) Feel free to ping me! 03:06, 14 October 2023 (UTC)

title: Talk:Arthur (TV series)


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Essential amino acid

An essential amino acid, or indispensable amino acid, is an amino acid that cannot be synthesized from scratch by the organism fast enough to supply its demand, and must therefore come from the diet. Of the 21 amino acids common to all life forms, the nine amino acids humans cannot synthesize are valine, isoleucine, leucine, methionine, phenylalanine, tryptophan, threonine, histidine, and lysine.

Six other amino acids are considered conditionally essential in the human diet, meaning their synthesis can be limited under special pathophysiological conditions, such as prematurity in the infant or individuals in severe catabolic distress. These six are arginine, cysteine, glycine, glutamine, proline, and tyrosine. Six amino acids are non-essential (dispensable) in humans, meaning they can be synthesized in sufficient quantities in the body. These six are alanine, aspartic acid, asparagine, glutamic acid, serine, and selenocysteine (considered the 21st amino acid). Pyrrolysine (considered the 22nd amino acid), which is proteinogenic only in certain microorganisms, is not used by and therefore non-essential for most organisms, including humans.

The limiting amino acid is the essential amino acid which is furthest from meeting nutritional requirements. This concept is important when determining the selection, number, and amount of foods to consume because even when total protein and all other essential amino acids are satisfied if the limiting amino acid is not satisfied then the meal is considered to be nutritionally limited by that amino acid.

Overview
(*) Pyrrolysine, sometimes considered the "22nd amino acid", is not used by the human body.

Essentiality in humans
Of the twenty amino acids common to all life forms (not counting selenocysteine), humans cannot synthesize nine: histidine, isoleucine, leucine, lysine, methionine, phenylalanine, threonine, tryptophan and valine. Additionally, the amino acids arginine, cysteine, glutamine, glycine, proline and tyrosine are considered conditionally essential, which means that specific populations who do not synthesize it in adequate amounts, such as newborn infants and people with diseased livers who are unable to synthesize cysteine, must obtain one or more of these conditionally essential amino acids from their diet. For example, enough arginine is synthesized by the urea cycle to meet the needs of an adult but perhaps not those of a growing child. Amino acids that must be obtained from the diet are called essential amino acids.

Eukaryotes can synthesize some of the amino acids from other substrates. Consequently, only a subset of the amino acids used in protein synthesis are essential nutrients.

From intermediates of the citric acid cycle and other pathways
Nonessential amino acids are produced in the body. The pathways for the synthesis of nonessential amino acids come from basic metabolic pathways. Glutamate dehydrogenase catalyzes the reductive amination of α-ketoglutarate to glutamate. A transamination reaction takes place in the synthesis of most amino acids. At this step, the chirality of the amino acid is established. Alanine and aspartate are synthesized by the transamination of pyruvate and oxaloacetate, respectively. Glutamine is synthesized from NH4+ and glutamate, and asparagine is synthesized similarly. Proline and arginine are both derived from glutamate. Serine, formed from 3-phosphoglycerate, which comes from glycolysis, is the precursor of glycine and cysteine. Tyrosine is synthesized by the hydroxylation of phenylalanine, which is an essential amino acid.

Recommended daily intake
Estimating the daily requirement for the indispensable amino acids has proven to be difficult; these numbers have undergone considerable revision over the last 20 years. The following table lists the recommended daily amounts currently in use for essential amino acids in adult humans (unless specified otherwise), together with their standard one-letter abbreviations.

The recommended daily intakes for children aged three years and older is 10% to 20% higher than adult levels and those for infants can be as much as 150% higher in the first year of life. Cysteine (or sulfur-containing amino acids), tyrosine (or aromatic amino acids), and arginine are always required by infants and growing children. Methionine and cysteine are grouped together because one of them can be synthesized from the other using the enzyme methionine S-methyltransferase and the catalyst methionine synthase. Phenylalanine and tyrosine are grouped together because one of them can be synthesized from the other using the enzyme phenylalanine/tyrosine ammonia-lyase.

Amino acid requirements and the amino acid content of food
Historically, amino acid requirements were determined by calculating the balance between dietary nitrogen intake and nitrogen excreted in the liquid and solid wastes, because proteins represent the largest nitrogen content in a body. A positive balance occurs when more nitrogen is consumed than is excreted, which indicates that some of the nitrogen is being used by the body to build proteins. A negative nitrogen balance occurs when more nitrogen is excreted than is consumed, which indicates that there is insufficient intake for the body to maintain its health. Graduate students at the University of Illinois were fed an artificial diet so that there was a slightly positive nitrogen balance. Then one amino acid was omitted and the nitrogen balance recorded. If a positive balance continued, then that amino acid was deemed not essential. If a negative balance occurred, then that amino acid was slowly restored until a slightly positive nitrogen balance stabilized and the minimum amount recorded.

A similar method was used to determine the protein content of foods. Test subjects were fed a diet containing no protein and the nitrogen losses recorded. During the first week or more there is a rapid loss of labile proteins. Once the nitrogen losses stabilize, this baseline is determined to be the minimum required for maintenance. Then the test subjects were fed a measured amount of the food being tested. The difference between the nitrogen in that food and the nitrogen losses above baseline was the amount the body retained to rebuild proteins. The amount of nitrogen retained divided by the total nitrogen intake is called net protein utilization. The amount of nitrogen retained divided by the (nitrogen intake minus nitrogen loss above baseline) is called biological value and is usually given as a percentage.

Modern techniques make use of ion exchange chromatography to determine the actual amino acid content of foods. The USDA used this technique in their own labs to determine the content of 7793 foods across 28 categories. The USDA published the final database in 2018 to the public.

The limiting amino acid depends on the human requirements and there are currently two sets of human requirements from authoritative sources: one published by WHO and the other published by USDA.

Protein quality
Various attempts have been made to express the "quality" or "value" of various kinds of protein. Measures include the biological value, net protein utilization, protein efficiency ratio, protein digestibility corrected amino acid score and the complete proteins concept. These concepts are important in the livestock industry, because the relative lack of one or more of the essential amino acids in animal feeds would have a limiting effect on growth and thus on feed conversion ratio. Thus, various feedstuffs may be fed in combination to increase net protein utilization, or a supplement of an individual amino acid (methionine, lysine, threonine, or tryptophan) can be added to the feed.

Protein per calorie
Protein content in foods is often measured in protein per serving rather than protein per calorie. For instance, the USDA lists 6 grams of protein per large whole egg (a 50-gram serving) rather than 84 mg of protein per calorie (71 calories total). For comparison, there are 2.8 grams of protein in a serving of raw broccoli (100 grams) or 82 mg of protein per calorie (34 calories total), or the Daily Value of 47.67g of protein after eating 1,690g of raw broccoli a day at 574 cal. An egg contains 12.5g of protein per 100g, but 4 mg more protein per calorie, or the protein DV after 381g of egg, which is 545 cal. The ratio of essential amino acids (the quality of protein) is not taken into account, one would actually need to eat more than 3 kg of broccoli a day to have a healthy protein profile, and almost 6 kg to get enough calories. It is recommended that adult humans obtain between 10–35% of their 2000 calories a day as protein.

Complete proteins in non-human animals
Scientists had known since the early 20th century that rats could not survive on a diet whose only protein source was zein, which comes from maize (corn), but recovered if they were fed casein from cow's milk. This led William Cumming Rose to the discovery of the essential amino acid threonine. Through manipulation of rodent diets, Rose was able to show that ten amino acids are essential for rats: lysine, tryptophan, histidine, phenylalanine, leucine, isoleucine, methionine, valine, and arginine, in addition to threonine. Rose's later work showed that eight amino acids are essential for adult human beings, with histidine also being essential for infants. Longer-term studies established histidine as also essential for adult humans.

Interchangeability
The distinction between essential and non-essential amino acids is somewhat unclear, as some amino acids can be produced from others. The sulfur-containing amino acids, methionine and homocysteine, can be converted into each other but neither can be synthesized de novo in humans. Likewise, cysteine can be made from homocysteine but cannot be synthesized on its own. So, for convenience, sulfur-containing amino acids are sometimes considered a single pool of nutritionally equivalent amino acids as are the aromatic amino acid pair, phenylalanine and tyrosine. Likewise arginine, ornithine, and citrulline, which are interconvertible by the urea cycle, are considered a single group.

Effects of deficiency
If one of the essential amino acids is not available in the required quantities, protein synthesis will be inhibited, irrespective of the availability of the other amino acids. Protein deficiency has been shown to affect all of the body's organs and many of its systems, for example affecting brain development in infants and young children; inhibiting upkeep of the immune system, increasing risk of infection; affecting gut mucosal function and permeability, thereby reducing absorption and increasing vulnerability to systemic disease; and impacting kidney function. The physical signs of protein deficiency include edema, failure to thrive in infants and children, poor musculature, dull skin, and thin and fragile hair. Biochemical changes reflecting protein deficiency include low serum albumin and low serum transferrin.

The amino acids that are essential in the human diet were established in a series of experiments led by William Cumming Rose. The experiments involved elemental diets to healthy male graduate students. These diets consisted of corn starch, sucrose, butterfat without protein, corn oil, inorganic salts, the known vitamins, a large brown "candy" made of liver extract flavored with peppermint oil (to supply any unknown vitamins), and mixtures of highly purified individual amino acids. The main outcome measure was nitrogen balance. Rose noted that the symptoms of nervousness, exhaustion, and dizziness were encountered to a greater or lesser extent whenever human subjects were deprived of an essential amino acid.

Essential amino acid deficiency should be distinguished from protein-energy malnutrition, which can manifest as marasmus or kwashiorkor. Kwashiorkor was once attributed to pure protein deficiency in individuals who were consuming enough calories ("sugar baby syndrome"). However, this theory has been challenged by the finding that there is no difference in the diets of children developing marasmus as opposed to kwashiorkor. Still, for instance in Dietary Reference Intakes (DRI) maintained by the USDA, lack of one or more of the essential amino acids is described as protein-energy malnutrition.

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

title: Essential amino acid


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Talk:Essential amino acid

Wiki Education Foundation-supported course assignment
This article was the subject of a Wiki Education Foundation-supported course assignment, between 27 January 2020 and 8 May 2020. Further details are available on the course page. Student editor(s): SeumMondal23.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 20:51, 16 January 2022 (UTC)

What happened?
I would have thought this subject is pretty important stuff regarding nutrition. I came here because limiting amino acid redirects here, yet nothing is mentioned about that subject. Going through the history of the article, I see there was a section on that subject way back in 2009, but it has been replaced by a text with inaccurate promotion of vegetarianism. I understand why a vegetarian might like to obfuscate such data, but that is probably a bad idea -you guys need protein too, irrespective if you get it from a complete source such as meat, eggs or milk, or if you get it by carefully combining different plant products. I don't really see the point of hiding information on limiting amino acids: if you eat rice and beans combined you basically have a complete protein source. The point is: if you eat meat, no problem, but if you eat only a single plant-based foodstuff like maize with nothing else, you will have deficiencies. Simple.

In order to repair this page I will re-add the info from 2009 and go through the vegetarian promotionalism to delete inaccuracies. Although I actually think "limiting amino acid" needs it's own page. Cheers, Leo Breman (talk) 19:26, 3 November 2019 (UTC)
 * How about providing a diff edit so we don't have to rummage to find something that was added/removed 10 years ago? I also couldn't find anything in the current article that qualifies as "inaccurate promotion of vegetarianism." Best to clarify your comments before making the proposed revisions. Rhode Island Red (talk) 20:45, 3 November 2019 (UTC)
 * Hi Rhode Island Red.

 * Diff edit: Yeah, shoulda done that.
 * My point: info about "limiting amino acid" had completely disappeared by 2010. Information about meat/milk/eggs being a complete protein had disappeared.
 * Inaccurate promotion of vegetarianism: If you check the edits I just finished making, and the edit comments, you will see what I am talking about. That section was off. I feel I repaired it now. As it now says again, accurately reflecting the sources which were already here, there is nothing unhealthy about eating only plants, as long as you combine foods.
 * I see there is something going round on internet now called the "complete protein myth" which is a popular myth among (some) vegetarians. This is misleading -it is true that all plants contain all amino acids to some degree, but the amino acid which is limiting is not present at sufficient quantities. Take maize: it is deficient in lysine which is limiting amino acid in this case, but cornmeal still contains 0.265 g per 100g, which is 13% of what you need. Ergo, you could eat 769g and you would reach you daily dietary intake, but it is misleading to claim it is therefore a food with a complete protein source. If you ate 500 gram of nshima a day you would eventually get sick and die, but if you ate the same amount of meat, or a combination of beans and maize, you wouldn't die (at least, not from protein deficiency). I hope you understand it now,  Leo Breman (talk) 21:47, 3 November 2019 (UTC)

 * Done with this now. Give it a few years and vegetarians will have screwed it up again, I'll warrant. I'm more used to calculating feed for other livestock, but when it comes down to it, calculating feed for humans is really the same. When vampires rule the earth and we're all in pens, a cost effective and nutritious fodder will be a tepid mash of a few hundred gram of mealy-meal, bit more than 100g soya, a pinch of salt, a few mealworms and green leaves thrown in for the B12 and C, and maybe some antibiotics; that should give a good FCR & fatten us up good for the harvest at ~16yrs when the growth curve flattens. Cheers,  Leo Breman (talk) 01:45, 4 November 2019 (UTC)
 * Thanks for noticing and cleaning up! DMacks (talk) 02:51, 4 November 2019 (UTC)

Essential amino acids in plants new paragraph needed
There is a suggestion in the article that only animal-based food can provide all essential amino acids (EAAs). That is not true and should be corrected. Most of the plants contain all EAAs. Looking at Swedish Food Agancy's "Amino acids in food" also checking ingredients at FoodData Central we can see that most of the plant-based food contain all EAAs. Although most of the plants have low EAAs content there are examples that soybeans, beans, nuts, quinoa are rich in EAAs. Soybeans meal contain actually more EAAs in total than beef, chicken or fish and it has the lowest sample food required to provide a total amount of EAAs equal to the recommended daily sum of total EAAs.

Furthermore:

Plant Foods Have a Complete Amino Acid Composition

"Soy, brown rice, pea, corn, and potato protein have essential amino acid contents that meet the requirements as recommended by the WHO/FAO/UNU (WHO/FAO/UNU Expert Consultation 2007) (Fig. 2). In addition, the essential amino acid content of potato protein (37%) is in fact greater when compared with casein (34%) and egg (32%). These data suggest that certain plant-based proteins could theoretically provide sufficient essential amino acids to allow a robust postprandial stimulation of muscle protein synthesis."

Well, I think the article needs improvement and update regarding EAAs in plant-based food.

Regards Guniarz (talk) 14:09, 5 January 2020 (UTC)

biochemistry
nonessential 112.79.96.23 (talk) 13:28, 29 March 2021 (UTC)

Is taurine really an amino acid?
This is the first time I have seen taurine included in a list of amino acids.

Taurine is referred to as an "amino acid" in the article. It is a common misconception that taurine is an amino acid according to https://sciencenotes.org/why-taurine-is-not-an-amino-acid/

"In addition, cysteine, tyrosine, and arginine are considered semiessential amino acids, and taurine a semiessential aminosulfonic acid in children. The metabolic pathways that synthesize these monomers are not fully developed." says https://en.wikipedia.org/wiki/Amino_acid in its *only* mention of taurine. So is taurine an aminosulfonic acid rather than an amino acid? https://en.wikipedia.org/wiki/Sulfamic_acid says, "Sulfamic acid, also known as amidosulfonic acid, amidosulfuric acid, aminosulfonic acid, sulphamic acid and sulfamidic acid, is a molecular compound with the formula H3NSO3." The article makes no mention of amino acids. So it seems that an aminosulfonic acid is not an amino acid. Polar Apposite (talk) 16:41, 21 December 2022 (UTC)

RDA of essential amino acids table misleading
US NAM figures for RDAs of EAAs/BCAAs are given on the referenced source in "mg/g of protein" rather than "mg/kg of body weight", which makes this table confusing to look at, I believe the US NAM figures all need to be multiplied by 0.8 (the amount of protein per kg weight that is recommended in that same paper), for the numbers to be in a consistent unit. Threeheadedpuppy (talk) 14:29, 18 February 2024 (UTC)

title: Talk:Essential amino acid


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Districts of Luxembourg

The three districts of Luxembourg (districts, Distrikte, Distrikter) were the top-level administrative divisions of the Grand Duchy of Luxembourg. The districts were further subdivided into cantons, which still exist:

 * 1) Diekirch District
 * 2) *Diekirch
 * 3) *Clervaux
 * 4) *Redange
 * 5) *Vianden
 * 6) *Wiltz
 * 7) Grevenmacher District
 * 8) *Grevenmacher
 * 9) *Echternach
 * 10) *Remich
 * 11) Luxembourg District
 * 12) *Luxembourg
 * 13) *Capellen
 * 14) *Esch-sur-Alzette
 * 15) *Mersch

The cantons were created on 24 February 1843. In 1857, Mersch District was created from the cantons of Mersch and Redange. However, this fourth district was abolished in 1867, when the re-arrangements of 1857 were undone.

The districts were abolished per 3 October 2015 leaving the Cantons, of which there are 12 as the most senior local authorities of Luxembourg

title: Districts of Luxembourg


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Talk:Linear independence

Projective space of dependencies
I removed the claim that the space of linear dependencies of a a set of vectors is a projectiv space, since I couldn't make sense of it. The space of linear dependencies is a plain vector space, the kernel of the obvious map from Kn to the vector space V. AxelBoldt 15:38 Oct 10, 2002 (UTC)

I've put that comment back, this time with an explanation. Michael Hardy 00:31 Jan 16, 2003 (UTC)

Vector or tuple?
The section "The projective space of linear dependences" currently starts with "A linear dependence among vectors v1, ..., vn is a vector (a1, ..., an)".

In colloquial speech and java "vector" is sometimes used for a tuple of varying length, but in mathematics a vector is an element of a vectorspace and not just any tuple. "A linear dependence [...] is a vector [...]", thus states that linear dependences form a vector space, which is not the case, as the 0 is missing.

Currently a linear dependence is called a vector just as v1 to vn, which can be misunderstood, that a linear dependence is a vector of the same vector space as v1 to vn, but it is not.

A linear dependence is, in the first place, a tuple, or more generally a family. It could have become a vector, if linear dependeces formed a vector space, but they don't, they form a projective space, which is a manifold and linear dependences are points of this manifold. Markus Schmaus 09:45, 15 Jun 2005 (UTC)

 * I do not agree that "A linear dependence [...] is a vector [...]" states that linear dependences form a vector space; it only states that they are elements of a vector space. Similarly, one could say "An eigenvector is a vector v such that ...", even though eigenvectors cannot be zero. -- Jitse Niesen 10:37, 15 Jun 2005 (UTC)

 * Do we agree that vector simply referes to an element of a vector space?

 * If so, we can rephrase both statements. "An eigenvector is an element of the vector space, such that ..." makes perfect sense, it is clear which vector space is meant. With "A linear dependence [...] is an element of the (a) vector space [...]" it is not clear what this vector space is, it is not the vector space of the vi-s, nor is it the vector space of linear dependences, as they form no vector space. We have to specify which vector space we mean and say "A linear dependence [...] is an element of the vector space of n-tuples [...]" or shorter "A linear dependence [...] is an n-tuple [...]".

 * Compare this with the beginning of quadratic function. "A quadratic function is a vector, ..." would be a bad beginning, as, even though polynomial functions form a vectorspace, a quadratic function is, in the first place, a polynomial function. Markus Schmaus 15:01, 15 Jun 2005 (UTC)

 * The article says "A linear dependence among vectors v1, ..., vn is a vector (a1, ..., an) [...]". It seems clear to me of which vector space the vector (a1, ..., an) is an element of, namely the vector space Kn. Furthermore, one needs the vector space structure to turn it into a projective space. However, I don't think it's worth it to argue about this, so change it to tuple if you insist. -- Jitse Niesen 16:01, 15 Jun 2005 (UTC)

 * Sorry for insisting, but there's an itching everytime I read the paragraph. Markus Schmaus 20:10, 19 Jun 2005 (UTC)

 * Please go ahead. Jitse Niesen 20:24, 19 Jun 2005 (UTC)

Linear indepence of sets?
The article currently starts with "a set of elements of a vector space is linearly independent [...]" I don't think this refers to a set in the mathematical sense. Suppose v1 = v2 = (1,0), the set { v1, v2 } is linearly independent as it contains only one element (1,0), which isn't zero. But the vectors v1 and v2 are not linearly independent, as v1 - v2 = 0. Linear independence is the property of a set, but the property of a family (mathematics). Markus Schmaus 15:37, 15 Jun 2005 (UTC)

 * The article currently starts with "a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set". If you take the set $$ S = \{ v_1, v_2 \} $$ with $$ v_1 = v_2 = (1,0)$$, then S has only one element, namely (1,0), and none of the elements of S can be written as a linear combination of other elements of S. -- Jitse Niesen 16:14, 15 Jun 2005 (UTC)

 * I have now been convinced by Markus Schmaus that there is indeed a problem here, see User talk:Markus Schmaus. Consider the following two statements:
 * A set of elements of a vector space is linearly independent if there is a nontrivial (finite) linear combination that gives zero;
 * A square matrix is invertible if and only if its columns are linearly independent.
 * Combining these statements gives: "A square matrix is invertible if and only if there is no nontrivial linear combination of the set of vectors, formed by the columns of the matrix." Now, apply this to the matrix
 * $$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}. $$
 * This matrix has two columns, which are equal, hence the set of vectors only has one element. This set is linearly independent, yet the matrix is not invertible; contradiction.

 * The question is now, how to resolve this problem? Possibilities are
 * Reformulate the definition as: "A family of elements of a vector space ..." This is Markus' preferred solution. I do not like it very much as it is not the standard definition in the English literature, and it uses the rather uncommon term family, but it might be the best solution.
 * Remove the statement "A square matrix is invertible if and only if its columns are linearly independent" in invertible matrix. There might be other statements lurking around that are similarly subtly wrong.
 * Have two definitions, one for finite-dimensional vector spaces ("The vectors $$v_1$$, ..., $$v_n$$ in some vector space ...") and one for infinite-dimensional spaces (in terms of families). This might be the most readable solution, but it is a bit ugly.
 * Do not care about it, since everybody knows what is meant.
 * I am not happy with any of these solutions, least of all with the last one, so I am soliciting comments from other contributors. -- Jitse Niesen 17:02, 17 Jun 2005 (UTC)

 * I don't think 4. is acceptable as people are looking on wikipedia, because they don't know what is meant. I'm often sloppy myself, but sloppiness is a luxury, we can't afford on wikipedia. Nor do I think it is an option to remove the "invertability" statement without replacement. It is true in some way and helps to understand the relation between linear independence, bases and matrices.

 * Currently I would suggest an initial paragraph starting with "A collection, more precisely a set, sequence, or family, of elements of a vectorspace …".

 * We will need two different formal definitions, as the current definition depends on a family, even though this is not mentioned explicitly. So there is even a subtle error within the same page.

 * If v1, v2, ..., vn are elements of V, we say that they are linearly dependent over K if there exist elements a1, a2, ..., an in K not all equal to zero such that:

 * $$ a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n \mathbf{v}_n = \mathbf{0}$$

 * If we pick v1 = v2 as it is the case with the matrix, a1 = 1 and a2 = -1 we get
 * v1 - v1 = 0
 * Hence v1, v2 are linearly dependent according to this definition, which is in accordance with the "invertibility" statement but not with the current initial paragraph. Markus Schmaus 15:39, 18 Jun 2005 (UTC)

a survey of English textbooks re: linear independence
There are 2 notions of linear independence, one whether you regard the collection of vectors as simply a set, the other whether you regard it as an indexed family. Each definition is necessary; e.g. without the "family" definition, the statements about column/row rank become false, whereas a statement such as "every vector space has a basis" has meaning with respect to the "set" definition. So, each is useful...and each should be included in the article.

Here is a survey of the most popular English language algebra textbooks:

 * Lang -- gets it right (almost...he's not 100% clear that a linear combination must "combine like terms") he makes clear that there are 2 definitions involved, and he gives the concrete example of v_1 = v_2 = ... = v_n to show where they differ. Hurray for Lang!
 * Dummit/Foote -- confuses the 2 definitions
 * Hungerford -- splits the difference, takes the set definition but requires the vectors to be unique
 * MacLane Birkhoff -- typically, translates the usual definitions into abstract nonsense, but essentially uses the family definition, except the emphasis is on finite dimensions, so I didn't immediately find general definition of basis, e.g.
 * Kurosh -- confuses the 2 definitions
 * Herstein (Topics) -- takes the set definition, but goes on proving things that should be stated with family definition
 * Hoffman/Kunze -- takes the set definition, but seems to avoid incorrect statements by using ordered bases and talking about rank instead of linearly independent columns/rows
 * Axler (Linear Algebra Done Right) -- takes the family definition, and dismisses the set definition (with explanation)! An exceptional case.

All in all, there's enough confusion to warrant:

 * 1) Giving both definitions.
 * 2) Explaining the difference, giving an example to show when they are not equal.
 * 3) Explain difference between family and ordered basis (family is indexed, but still unordered)
 * 4) Show when and exactly what way the family definition is or is not needed.

Revolver 16:54, 19 Jun 2005 (UTC)

''Have two definitions, one for finite-dimensional vector spaces ("The vectors v_1, ..., v_n in some vector space ...") and one for infinite-dimensional spaces (in terms of families). This might be the most readable solution, but it is a bit ugly.''
 * This doesn't address the issue. The problem remains, whether one restricts to finite dimensional vector spaces or not. Revolver 16:59, 19 Jun 2005 (UTC)

 * Wow, I never thought that it would be so hard to define linear independence. I am now convinced that we need to include the definition using families. I tried to change the definition in the article to accommodate this discussion. Note that I did not change the very first sentence of the article, since I feel it is okay to be slightly sloppy there. I did not write anything about the differences between the family-based and set-based definition; I do not want to spend too much space on it, but we should probably write a few lines about it. Let me know what you think about it. -- Jitse Niesen 18:39, 19 Jun 2005 (UTC)

 * In that case I'd prefer collection instead of set, as the first can also refer to a family. Markus Schmaus 19:52, 19 Jun 2005 (UTC)

 * Thanks for the survey on English text books. Markus Schmaus 19:52, 19 Jun 2005 (UTC)

 * Revolver, I forgot to do say so, but also thanks from me.
 * Markus, regarding collection versus set, be bold! (I didn't know that collection can also refer to a family). By the way, thanks for catching my embarrassing mistake in the rewrite of the definition. Jitse Niesen 20:24, 19 Jun 2005 (UTC)
 * I don't think collection is a precise mathematical term, it's kind of soft, sometimes refering to a set, somtimes to a class, sometimes to a family, and maybe to other things I don't know. There is a pragraph on collection defining it as equivalent to a set but I don't think that's common usage, so I'm going to change that. Markus Schmaus

 * A term that may be more accessible: Michael Artin's Algebra (ISBN 0130047635) uses "ordered set". I've also seen "list", defined as a mapping from the natural numbers, but that would imply countability. Artin also states a finite-dimensional form, then expands, as suggested above. (Bourbaki might sneer, but students won't.) I believe the infinite-dimensional case is too important to omit in the body, though perhaps the opening can acknowledge it without a definition. Incidentally, Artin, and Mac Lane&Birkhoff (ISBN 0828403309, not the older Survey), take care to say that for some uses a basis can be a set (unordered), but most practical work requires the ordering. --KSmrqT 03:40, 4 November 2005 (UTC)

calculus not required
I can do that calculus example without calculus: divide both sides by e^t. then you have a be^t = constant, which is only true when b=0. Why use calculus if we don't need it? I'm going to change it. -Lethe | Talk 01:10, August 9, 2005 (UTC)

examples
I removed the calculus from the proof of example III. I also think that there are too many uninteresting examples. Example I and II are almost identical, and example III isn't too interesting either. what about an example of an infinite family of independent vectors, like say a basis for l^2. -Lethe | Talk 01:18, August 9, 2005 (UTC)

Experiment
Just thinking aloud: Is it an improvement to change the current opening from this:  In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. For instance, in three-dimensional Euclidean space R3, the three vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1) are linearly independent, while (2, −1, 1), (1, 0, 1) and (3, −1, 2) are not (since the third vector is the sum of the first two). Vectors which are not linearly independent are called linearly dependent.  to something like this:  In linear algebra, an ordered set of finite-dimensional vectors is linearly independent if no vector in the set can be written as a linear combination (a weighted sum) of those preceding it. That is, no vector is in the span of its predecessors. More broadly, in a possibly infinite-dimensional vector space, a family of vectors is linearly independent if any vector in the full space can be written in at most one way as a linear combination of a finite number of those in the family. Vectors which are not linearly independent are called linearly dependent.

\begin{matrix} \mbox{independent}\qquad\\ \underbrace{ \overbrace{ \begin{bmatrix}0\\0\\1\end{bmatrix}, \begin{bmatrix}0\\2\\-2\end{bmatrix}, \begin{bmatrix}1\\-2\\1\end{bmatrix} }, \begin{bmatrix}4\\2\\3\end{bmatrix} }\\ \mbox{dependent}\\ \end{matrix} $$  I see both advantages and disadvantages. Mainly, I thought it might help to experiment, to free our thinking a little. --KSmrqT 08:13, 4 November 2005 (UTC)
 * Vector spaces, not vectors, have dimensionality. Therefore the phrase "finite-dimensional vectors" has no meaning.  Not too keen on predecessors.  Prefer family to ordered set. -Lethe | Talk 10:36, 4 November 2005 (UTC)
 * I think it is fine the way it is - ordering isn't necessary. On a different matter, should the projective spaces of linear dependencies bit really be before the examples of linear independence? JPD (talk) 10:50, 4 November 2005 (UTC)
 * (JPD, if you have no interest in my question, please take up yours in a different thread, not this one. Thanks.)
 * Yes, it might be prudent to say "an ordered set of vectors in a finite-dimensional vector space", though the shorthand is often used for convenience. One of my questions is whether order is an advantage or disadvantage. Or is it even better to have both a "procedural" definition and a "universal" definition (bordering on category theory), as illustrated. Lethe, do you want to defer mention of order to the article body, or omit it entirely? And should I assume that you like the "at most one way" definition (borrowed from Mac Lane&Birkhoff)?
 * I have one definite opinion: The column vector display of independence and dependence works much better than the inline version. (Also, I do think it best to choose vectors that are not orthonormal.) --KSmrqT 12:33, 4 November 2005 (UTC)
 * I take it that you actually think I should discuss the ordering in a separate thread, even though I am interested in your question. That's a fair point. I agree that the column vector display is more illuminating, but I don't particularly like any of your other suggested changes. JPD (talk) 13:56, 4 November 2005 (UTC)
 * I was objecting to "On a different matter…". Order is fair game. So far, no fans. Which is interesting in itself, because it is no more restrictive (in finite dimensions) than unordered, but more practical. It's hard to tell from the brief responses whether the antipathy is a matter of taste or technical difficulties (not that there's a sharp line). But, unless someone speaks up it looks like "family" — despite being unfamiliar to almost anyone needing to learn about linear independence — is here to stay. I'm not surprised, and I can live with "family"; but I am a little disappointed that we can't seem to do better. --KSmrqT 21:07, 4 November 2005 (UTC)
 * I'm not one hundred percent sure, but I think in my case, my complaints are purely matters of taste. -Lethe | Talk 00:27, 5 November 2005 (UTC)
 * I guess you could say my objections are taste as well, but I think it's important that linear dependence does not in any way depend on order. I don't see how the order makes it any more practical. Earlier, someone suggested "collection" rather than family. I guess the point is that linear independence can be defined for sets, or sequences (finite or infinite), in exactly the same way. I doubt that anyone who doesn't already know about linear independence is going to be bothered about whether define it for a set, sequence, family or collection. JPD (talk) 14:45, 5 November 2005 (UTC)
 * My reading of previous discussion is that "family" is preferred over "collection" (too vague, per Markus Schmaus) and "set" (unacceptable, because a set excludes duplicates). In the finite-dimensional case, "list", a mapping from natural numbers to items (here, vectors), is OK; but we can have an uncountable collection in an infinite-dimensional case.
 * On order, three comments:
 * In saying one vector is a linear combination of others, we already introduce an artificial asymmetry, if not an order per se. If three vectors are dependent, which is the culprit? The "at most one" definition has no such asymmetry. The asymmetric version also hides a subtle flaw: the zero vector should not be considered independent, even if it is the only vector given.
 * In giving a first definition using order, we do not restrict; it only seems so. For, we can prove that if a list is dependent, so is any reordering of it. This should appear explicitly in the body, because even the family definition currently used does not make this explicit.
 * Stipulate that the empty set is linearly independent, and that its span is the zero space consisting of just the zero vector. Many practical computations involving linear independence, and certainly "basis", will actually use an ordering. For example, Gram-Schmidt orthogonalization in an inner product space converts an independent list into an orthonormal list. Or consider the theorem that if an ordered set spans V then it contains a linearly independent subset that spans V. So whether we start with an ordered definition or not, we will eventually want to derive and use one.
 * Incidentally, may I assume that nobody likes the "at most one" definition either? --KSmrqT 22:18, 5 November 2005 (UTC)
 * I think I see what you mean about the introduced assymetry, for that sort of reason would agree that "one vector is a linear combination of the others" is not my favourite definition of dependence. However, it is easier to grasp for most people, so I was not about to change it. I do not think there is actually any assymmetry there, especially when defining independence rather than dependence. I think the current version and use cover the zero vector validly and equally well, but it might be worth adding a sentence to make it clear.
 * I agree that we don't restrict by using order, by I don't see any reason to use order in the definition and then show that it doesn't matter, when there are perfectly good definitions which don't use order at all. The family doesn't need to make make the irrelevance of reording explicit, because there is no explicit ordering in th e concept of a family.
 * I agree that the empty set should be mentioned in the body in some way. I don't think the fact that we often use ordered bases is relevant. The ordering is not part of the definition of basis, and definitely not needed for linear independence. The general definition covers the ordered case, so we do not need to derive a definition, in order to have linear indepedence in an ordered set or a sequence. Lastly, collection is a bit vague, but that is not a completely bad thing. Family is good because it is easily understood in the same vague sense by those who don't want detail, yet does have a detailed definition for those of us who care. JPD (talk) 19:56, 6 November 2005 (UTC)

Linear dependences section
It seems a bit strange that this section is before the examples. Does anyone agree? JPD (talk) 13:56, 4 November 2005 (UTC)

 * Yes. If I was teaching students I definitely wouldn't do it in that order!  --RFBailey 21:04, 6 November 2005 (UTC)

 * A strong alternative is to first define a linear relationship or dependency or whatever as a weighted sum of vectors, with not all weights zero, that equals the zero vector. Then define linear independence as the lack of a linear relationship among any of the vectors. That helps several weaknesses at once:
 * remove asymmetry from definition
 * shows a set with only the zero vector is not independent
 * better integrates the digression about projective space
 * supports definition of mapping from scalar tuples to vectors, Kn → V, defined by any n-family in V, which is
 * monomorphism for independent family
 * epimorphism for spanning family
 * isomorphism for basis family, thus defining coordinates
 * Otherwise, the whole discussion of projective space seems unmotivated. --KSmrqT 05:05, 7 November 2005 (UTC)

Firstly, however indepedence or dependence are defined in the article, I think the examples should come before any mention of the projective space of linear dependences. Apart from that, I agree that KSmrq's suggestion would be a good approach in the Definition section and following. I don't see where the where some of the weaknesses he mentions are, but it would be good to make this section flow better and give a bit more idea of what the relevance of linear independence is. JPD (talk) 15:56, 7 November 2005 (UTC)

Linear independence over rings
It should at least be mentioned that in a module over a ring, linear independence (in the sense that zero cannot be represented as a non-trivial linear combination) is not equivalent to one vector being contained in the span of the remaining elements.--80.136.131.201 18:37, 20 January 2006 (UTC)
 * This article is only about linear independence over the real numbers. For general rings, things are messy indeed, and I don't even think that the concept if linear independence is at all useful. Oleg Alexandrov (talk) 23:45, 20 January 2006 (UTC)
 * Some parts of the usual theory work pretty well: Linear independence of the columns of a matrix corresponds to uniqueness of solutions (if any exist), a subset of a module is a basis iff it is linearly independent and generating. Cf. the characterization based on properties of the induced map $$K^n\to V$$ mentioned by KSmrq in the preceding section.--80.136.131.201 00:14, 21 January 2006 (UTC)

determinant check
The article states that a check of independence can use the determinant when the number of vectors equals the dimension. In effect, this is a check to see if the vectors form a basis. But even when the number of vectors, m, is less than the dimension, n, we can check for independence by requiring all m×m subdeterminants to be nonzero. For example, the following three vectors of dimension four are independent.

 * $$v_1 = \begin{bmatrix}4\\5\\6\\7\end{bmatrix},\quad v_2 = \begin{bmatrix}2\\3\\5\\8\end{bmatrix},\quad v_3 = \begin{bmatrix}1\\1\\2\\3\end{bmatrix}$$

 * $$\begin{vmatrix}4&2&1\\5&3&1\\6&5&2\end{vmatrix} \ne 0,\quad \begin{vmatrix}4&2&1\\5&3&1\\7&8&3\end{vmatrix} \ne 0,\quad \begin{vmatrix}4&2&1\\6&5&2\\7&8&3\end{vmatrix} \ne 0,\quad \begin{vmatrix}5&3&1\\6&5&2\\7&8&3\end{vmatrix} \ne 0$$

I'd suggest this is worth mentioning in the article. --KSmrqT 01:15, 31 January 2006 (UTC)
 * Sure, put it in. Personally, I always like to err on the side of exhaustiveness, this being an encyclopedia, not a textbook.  By the way, both statements are implied by the more high-brow statement that a set of vectors is dependent iff their wedge product is 0.  -lethe talk + 01:54, 31 January 2006 (UTC)

 * Shouldn't that be 'by requiring at least one of the sub-determinants to be non-zero'? 137.205.139.149 20:58, 19 January 2007 (UTC)

 * You're a year late to the party. The article has already been augmented, and you can check that the language used there is correct. It says we have dependence (not independence) if all subdeterminants are zero, which (as you correctly observe) means that at least one nonzero subdeterminant indicates independence. But as a practical matter, for numerical computation we would use a rank-revealing decomposition. --KSmrqT 17:12, 20 January 2007 (UTC)

 * I disagree. I was reading the article and it wasn't clear to me whether it was required that all determinants be zero or one of them being zero was enough to have dependence. Maybe the language is correct, but it is not clear enough for those of us trying to learn. — Preceding unsigned comment added by Viviannevilar (talk • contribs) 02:14, 22 June 2011 (UTC)

Can this be proven simply (ie. without involving wedge product and in an uncomplicated fashion)? If so, it would probably be worthwhile to reference some such proof here. Robryk (talk) 22:50, 25 January 2008 (UTC)

Example in lead
Did anyone actually solve the first example? I think it should be (9,2,3) instead of (4,2,3). —Preceding unsigned comment added by Caelumluna (talk • contribs) 02:38, 20 September 2007 (UTC)

 * It looks fine to me. -- Jitse Niesen (talk) 04:30, 20 September 2007 (UTC)

New section
I've removed the newly-added "Formula" section, because it's overly-complicated, and more importantly, redundant.

The article already states that a set of vectors is linearly independent iff the determinant of the matrix they form (let's call it X) is non-zero. However, the matrix formed in the new section (the matrix of dot products) is in fact equivalent to calculating XTX. Of course, |XTX| ≠ 0 iff |X| ≠ 0. Therefore, this section says nothing new, I'm afraid. Oli Filth(talk) 23:25, 8 May 2008 (UTC)

 * OK, I basically agree, but it is not that trivial: suppose you have a set of 4 vectors which 'live' in $$\mathbb{R}^7$$ and you want to find out if they are linearly independent or not. Then matrix X will have 4 columns and 7 rows: it is not square, so $$\det X $$ is undefined.  In such case it is not a redundancy to compute $$\det X^T X$$.  And then, in such case (for non-square X), how does one show (succintly) that $$\det X^T X$$ tests for linear independence?  Here is an argument: the 4-D subspace $$\{\mathbf{x}_1,...,\mathbf{x}_4\}$$ contains some orthonormal basis $$\{\mathbf{e}_1,...,\mathbf{e}_k\}$$ where $$k \le 4$$.  Arrange these vectors in columns to form matrix E, which will be 7 × k.  Then $$ E^T $$ is k × 7 and a left-inverse of E, so $$ E^T E = I $$ and $$ E E^T $$ is a 7 × 7 symmetric projection operator which projects vectors in $$\mathbb{R}^7$$ onto the k-dimensional subspace.  As such, let $$\Pi = E E^T$$.  Then $$\Pi X = X$$ and so $$ X^T \Pi X = X^T (\Pi X) = X^T X = (X^T E) (E^T X) = (E^T X)^T (E^T X) = X_{(E)}^T X_{(E)} $$ where $$ X_{(E)}^T X_{(E)} $$ is a 4 × 4 square matrix and $$X_{(E)} := E^T X $$ is the matrix X expressed in terms of the basis $$\{\mathbf{e}_1,...,\mathbf{e}_k\}$$.  Now it is indeed true${\color{Blue} {}^1}$ that $$ \det X_{(E)} = \sqrt{\det X_{(E)}^T X_{(E)} } $$ where the left side is the usual test for linear independence (as well as giving the 4-volume of the 'parallelotope' whose sides are parallel to $$\{\mathbf{x}_{1(E)},...,\mathbf{x}_{4(E)}\}$$).  [$${\color{Blue}1}$$ - because $$ \det X_{(E)}^T = \det X_{(E)} $$, so $$ \det (X_{(E)}^T X_{(E)}) = (\det X_{(E)}^T) (\det X_{(E)}) = (\det X_{(E)})^2 $$.  But in such cases (of non-square X) it is easier to compute $$ \det X^T X $$ directly rather than first finding E (say, by Gram-Schmidt orthogonalization) and then computing $$X_{(E)} := E^T X $$ and then $$ \det X_{(E)} $$.]

 * Note that since $$ X^T X = X_{(E)}^T X_{(E)} $$, that is, $$ X^T X $$ is invariant under change of (the subspace's) basis, then $$ X^T X $$ would be a tensor, not unlike the metric tensor.

 * Anyway, today I found out that $$ X^T X $$ is called Gram matrix, so I added a See also link to it, and that should be enough. There is a nice PDF file on the web about this topic: http://www.owlnet.rice.edu/~fjones/chap8.pdf, followed up with http://www.owlnet.rice.edu/~fjones/chap11.pdf, though this goes well beyond mere linear independence (perhaps I'll add the first one under External links for the Gram matrix article).  Anyway, thank you for describing my edit as 'good faith'.  —AugPi (talk) 02:10, 10 May 2008 (UTC)

 * You are absolutely correct that det(X) only exists for square matrices. I realised my mistake as soon as I had performed my edits, and was about to update the existing material on determinants to also discuss det(XTX), but then got distracted.  I will make these changes today!  Oli Filth(talk) 11:24, 10 May 2008 (UTC)

algebraic numbers
The page on Lindemann–Weierstrass theorem mentions algebraic numbers that are linearly independent over the rationals. I assume this is a similar idea as with vectors, i.e. a set of algebraic numbers are linearly independent if none can be expressed as a (finite) sum of any other number or numbers in the set, or a product of any set-member with a rational. Is this an extension of the concept of linear independence that should be mentioned in the article? --Maltelauridsbrigge (talk) 14:26, 6 August 2008 (UTC)

 * In algebraic number theory you sometimes think of number fields as being vector spaces over the field Q, the fact that your vectors are also numbers is odd but mathematically fine. So algebraic numbers ai are linearly independent over Q if the equation
 * $$c_1 a_{i_1}+\ldots+c_n a_{i_n}=0$$
 * with rational numbers c1,...,cn and distinct ai j only ever has the solution c1=...=cn=0. So this isn't really an extension of the ideas discussed in the article, it's just applying them to a vector space that doesn't look like the ones you first learn about, like R2 and R3. Chenxlee (talk) 20:58, 3 February 2009 (UTC)

Inaccuracy in first line?
Hi, I could be wrong here but I think the first line in the article needs to be rewritten. It reads:

In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the family

Strictly speaking, shouldn't it instead read something like ...if no linear combination of finitely many vectors in the family can equal zero.

In a vector space over Z2, for example, (2,0) and (3,0) would incorrectly be classed as linearly independent by the first definition, since neither is an integer multiple of the other. — Preceding unsigned comment added by Insperatum (talk • contribs) 00:41, 17 October 2012 (UTC)

 * There is no vector space over Z, the integers, because Z is not a field. Your suggested wording introduces complications (and the existing wording does not have the problem you suggest for the reason I've given): the sum of a no vectors (zero is a finite number) equals zero, making every set "linearly dependent" by your definition. — Quondum 04:18, 17 October 2012 (UTC)

 * Hi. the first line at the moment "In the theory of vector spaces, a set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent. " doesn't make it apparent that {0} is a linear dependent set as there are no other vectors in the set. — Preceding unsigned comment added by 146.179.206.233 (talk) 12:42, 25 November 2016 (UTC)

Evaluating Linear Independence
It seems to me that the algebra is incorrect in the $R^{2}$ section (the 2-dimensional section). The row reduction should not create a unit vector... it looks like a math error. But I didn't want to change it, in case I'm wrong. Can someone verify that it's incorrect? (It should give a matrix with the first column as [1, 0], which is correct, but the second column should be [0, 5], shouldn't it?

This.is.mvw (talk) 22:42, 1 August 2017 (UTC)
 * I see nothing wrong with the computation as it is given. The algebraic steps are exactly those given above in the three vector case. The second columns are respectively, [-3, 2], [-3, 5], [-3, 1], and [0, 1]. --Bill Cherowitzo (talk) 02:44, 2 August 2017 (UTC)

 * I find it interesting that the latest revision is designed to include a definition for the linear dependence of sets consisting of zero and one vectors. Can the empty set of vectors really be contemplated as being linearly independent or linearly dependent?  I think it is even difficult to view a single vector as linearly dependent or independent on its own.Prof McCarthy (talk) 22:41, 4 September 2018 (UTC)

title: Talk:Linear independence


text:
Net protein utilization

The net protein utilization, or NPU, is the percentage of ingested nitrogen that is retained in the body. It is used to determine the nutritional efficiency of protein in the diet, that is, it is used as a measure of "protein quality" for human nutritional purposes.

As a value, NPU can range from 0 to 1 (or 100), with a value of 1 (or 100) indicating 100% utilization of dietary nitrogen as protein and a value of 0 an indication that none of the nitrogen supplied was converted to protein.

Certain foodstuffs, such as eggs or milk, rate as 1 on an NPU chart.

Experimentally, this value can be determined by determining dietary protein intake and then measuring nitrogen excretion. One formula for apparent NPU is:

 * NPU = {0.16 × (24 hour protein intake in grams)} - {(24 hour urinary urea nitrogen) + 2} - {0.1 × (ideal body weight in kilograms)} / {0.16 × (24 hour protein intake in grams)}

NPU and biological value (BV) both measure nitrogen retention; the difference is that biological value is calculated from nitrogen absorbed, whereas net protein utilization is from nitrogen ingested. Another closely related quantity is the net postprandial protein utilization (NPPU), which is the maximum potential NPU of a dietary protein source under ideal conditions.

The Protein Digestibility Corrected Amino Acid Score (PDCAAS) is a more modern rating for determining protein quality, and the current ranking standard used by the FDA.

The Digestible Indispensable Amino Acid Score (DIAAS) is a protein quality method, proposed in March 2013 by the Food and Agriculture Organization to replace the current protein ranking standard, the Protein Digestibility Corrected Amino Acid Score (PDCAAS). The proposition is contested, however, due to lack of data.

title: Net protein utilization


text:
Orthogonality

In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.

Orthogonality is also used with various meanings that are often weakly related or not related at all with the mathematical meanings.

Etymology
The word comes from the Ancient Greek ('), meaning "upright", and  ('), meaning "angle".

The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle.

Optics
In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization.

Special relativity
In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simultaneous events, also determined by the rapidity. The theory features relativity of simultaneity.

Quantum mechanics
In quantum mechanics, a sufficient (but not necessary) condition that two eigenstates of a Hermitian operator, $$ \psi_m $$ and $$ \psi_n $$, are orthogonal is that they correspond to different eigenvalues. This means, in Dirac notation, that $$ \langle \psi_m | \psi_n \rangle = 0 $$ if $$ \psi_m $$ and $$ \psi_n $$ correspond to different eigenvalues. This follows from the fact that Schrödinger's equation is a Sturm–Liouville equation (in Schrödinger's formulation) or that observables are given by Hermitian operators (in Heisenberg's formulation).

Art
In art, the perspective (imaginary) lines pointing to the vanishing point are referred to as "orthogonal lines". The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as Piet Mondrian and Burgoyne Diller are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines that are straight and exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay at the web site of the Thyssen-Bornemisza Museum states that "Mondrian ... dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours."

Computer science
Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results. This usage was introduced by Van Wijngaarden in the design of Algol 68:

The number of independent primitive concepts has been minimized in order that the language be easy to describe, to learn, and to implement. On the other hand, these concepts have been applied “orthogonally” in order to maximize the expressive power of the language while trying to avoid deleterious superfluities.

Orthogonality is a system design property which guarantees that modifying the technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. Typically this is achieved through the separation of concerns and encapsulation, and it is essential for feasible and compact designs of complex systems. The emergent behavior of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration, i.e., non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it is easier to verify designs that neither cause side effects nor depend on them.

Orthogonal instruction set
An instruction set is said to be orthogonal if it lacks redundancy (i.e., there is only a single instruction that can be used to accomplish a given task) and is designed such that instructions can use any register in any addressing mode. This terminology results from considering an instruction as a vector whose components are the instruction fields. One field identifies the registers to be operated upon and another specifies the addressing mode. An orthogonal instruction set uniquely encodes all combinations of registers and addressing modes.

Telecommunications
In telecommunications, multiple access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different basis functions. One such scheme is time-division multiple access (TDMA), where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").

Orthogonal frequency-division multiplexing
Another scheme is orthogonal frequency-division multiplexing (OFDM), which refers to the use, by a single transmitter, of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (a, g, and n) versions of 802.11 Wi-Fi; WiMAX; ITU-T G.hn, DVB-T, the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT (Discrete Multi Tone), the standard form of ADSL.

In OFDM, the subcarrier frequencies are chosen so that the subcarriers are orthogonal to each other, meaning that crosstalk between the subchannels is eliminated and intercarrier guard bands are not required. This greatly simplifies the design of both the transmitter and the receiver. In conventional FDM, a separate filter for each subchannel is required.

Statistics, econometrics, and economics
When performing statistical analysis, independent variables that affect a particular dependent variable are said to be orthogonal if they are uncorrelated, since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with simple regression or simultaneously with multiple regression. If correlation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting the expected value (the mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e., vectors) and as random variables (i.e., density functions). One econometric formalism that is alternative to the maximum likelihood framework, the Generalized Method of Moments, relies on orthogonality conditions. In particular, the Ordinary Least Squares estimator may be easily derived from an orthogonality condition between the explanatory variables and model residuals.

Taxonomy
In taxonomy, an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive.

Chemistry and biochemistry
In chemistry and biochemistry, an orthogonal interaction occurs when there are two pairs of substances and each substance can interact with their respective partner, but does not interact with either substance of the other pair. For example, DNA has two orthogonal pairs: cytosine and guanine form a base-pair, and adenine and thymine form another base-pair, but other base-pair combinations are strongly disfavored. As a chemical example, tetrazine reacts with transcyclooctene and azide reacts with cyclooctyne without any cross-reaction, so these are mutually orthogonal reactions, and so, can be performed simultaneously and selectively.

Organic synthesis
In organic synthesis, orthogonal protection is a strategy allowing the deprotection of functional groups independently of each other.

Supramolecular chemistry
In supramolecular chemistry the notion of orthogonality refers to the possibility of two or more supramolecular, often non-covalent, interactions being compatible; reversibly forming without interference from the other.

Analytical chemistry
In analytical chemistry, analyses are "orthogonal" if they make a measurement or identification in completely different ways, thus increasing the reliability of the measurement. Orthogonal testing thus can be viewed as "cross-checking" of results, and the "cross" notion corresponds to the etymologic origin of orthogonality. Orthogonal testing is often required as a part of a new drug application.

System reliability
In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup device or method is completely different from the prone to error device or method. The failure mode of an orthogonally redundant back-up device or method does not intersect with and is completely different from the failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic failure.

Neuroscience
In neuroscience, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map.

Philosophy
In philosophy, two topics, authors, or pieces of writing are said to be "orthogonal" to each other when they do not substantively cover what could be considered potentially overlapping or competing claims. Thus, texts in philosophy can either support and complement one another, they can offer competing explanations or systems, or they can be orthogonal to each other in cases where the scope, content, and purpose of the pieces of writing are entirely unrelated.

Gaming
In board games such as chess which feature a grid of squares, 'orthogonal' is used to mean "in the same row/'rank' or column/'file'". This is the counterpart to squares which are "diagonally adjacent". In the ancient Chinese board game Go a player can capture the stones of an opponent by occupying all orthogonally adjacent points.

Other examples
Stereo vinyl records encode both the left and right stereo channels in a single groove. The V-shaped groove in the vinyl has walls that are 90 degrees to each other, with variations in each wall separately encoding one of the two analogue channels that make up the stereo signal. The cartridge senses the motion of the stylus following the groove in two orthogonal directions: 45 degrees from vertical to either side. A pure horizontal motion corresponds to a mono signal, equivalent to a stereo signal in which both channels carry identical (in-phase) signals.

title: Orthogonality


text:
User:Elian/POV

As I have already stated, I am biased pro-palestinian. That doesn't mean I am uncritical to this side, but my sympathies tend to what I regard as the weaker side in the conflict. My stand may vary depending on with whom I am discussing - if I see, as in wikipedia, one side nearly unrepresented, it's natural for me to take this side, but you could also find me as a vehement supporter of Israel's right of existence in a discussion with Arabs.

Beside, I am an adherent of a thing called "Realpolitik", meaning that political solutions should be oriented towards the feasible and not always insist on "what is right and just". However, I can be very adamant if someone tries to tell me that killing civilians "is the right thing to do" because there is "no other solution". This is a distinction I miss in many articles on the conflict which often tend to mix these things.

back to Arab-Israeli conflict editing project

title: User:Elian/POV


text:
SuperH

SuperH (or SH) is a 32-bit reduced instruction set computing (RISC) instruction set architecture (ISA) developed by Hitachi and currently produced by Renesas. It is implemented by microcontrollers and microprocessors for embedded systems.

At the time of introduction, SuperH was notable for having fixed-length 16-bit instructions in spite of its 32-bit architecture. Using smaller instructions had consequences: the register file was smaller and instructions were generally two-operand format. However for the market the SuperH was aimed at, this was a small price to pay for the improved memory and processor cache efficiency.

Later versions of the design, starting with SH-5, included both 16- and 32-bit instructions, with the 16-bit versions mapping onto the 32-bit version inside the CPU. This allowed the machine code to continue using the shorter instructions to save memory, while not demanding the amount of instruction decoding logic needed if they were completely separate instructions. This concept is now known as a compressed instruction set and is also used by other companies, the most notable example being ARM for its Thumb instruction set.

In 2015, many of the original patents for the SuperH architecture expired and the SH-2 CPU was reimplemented as open source hardware under the name J2.

SH-1 and SH-2
The SuperH processor core family was first developed by Hitachi in the early 1990s. The design concept was for a single instruction set (ISA) that would be upward compatible across a series of CPU cores.

In the past, this sort of design problem would have been solved using microcode, with the low-end models in the series performing non-implemented instructions as a series of more basic instructions. For instance, a "long multiply" (multiplying two 32-bit registers to produce a 64-bit product) might be implemented in hardware on high-end models but instead be performed as a series of additions on low-end models.

One of the key realizations during the development of the RISC concept was that the microcode had a finite decoding time, and as processors became faster, this represented an unacceptable performance overhead. To address this, Hitachi instead developed a single ISA for the entire line, with unsupported instructions causing traps on those implementations that didn't include hardware support. For instance, the initial models in the line, the SH-1 and SH-2, differed only in their support for 64-bit multiplication; the SH-2 supported MUL, DMULS and DMULU, whereas the SH-1 would cause a trap if these were encountered.

The SH-1 was the basic model, supporting a total of 56 instructions. The SH-2 added 64-bit multiplication and a few additional commands for branching and other duties, bringing the total to 62 supported instructions. The SH-1 and the SH-2 were used in the Sega Saturn, Sega 32X and Capcom CPS-3.

The ISA uses 16-bit instructions for better code density than 32-bit instructions, which was important at the time due to the high cost of main memory and the implementation cost of cache. As of 2023, code density is still important for small embedded systems and massively multicore processors. The downsides to this approach were that there were fewer bits available to encode a register number or a constant value. In the original SuperH ISA, there were only 16 general registers, requiring four bits for the source and another four for the destination; however some instructions have an implied R0, R15, or a system register as an extra operand. The instruction opcode is four, eight, twelve, or sixteen bits long, and the remaining four-bit fields are used for register or immediate operands in various ways: there are twelve classes of instructions, for a total of 142 instructions in SH-2.

Delayed branches are introduced for both SH-1 and SH-2. Unconditional branch instructions have one delay slot.

SH-3
A few years later, the SH-3 core was added to the family; new features included another interrupt concept, a memory management unit (MMU), and a modified cache concept. These features required an extended instruction set, adding six new instructions for a total of 68. The SH-3 was bi-endian, running in either big-endian or little-endian byte ordering.

The SH-3 core also added a DSP extension, then called SH-3-DSP. With extended data paths for efficient DSP processing, special accumulators and a dedicated MAC-type DSP engine, this core unified the DSP and the RISC processor world. A derivative of the DSP was also used with the original SH-2 core.

Between 1994 and 1996, 35.1 million SuperH devices were shipped worldwide.

SH-4
In 1997, Hitachi and STMicroelectronics (STM) started collaborating on the design of the SH-4 for the Dreamcast. SH-4 featured superscalar (2-way) instruction execution and a vector floating-point unit (particularly suited to 3D graphics). Standard chips based on the SH-4 were introduced around 1998.

Licensing
In early 2001, Hitachi and STM formed the IP company SuperH, Inc., which was going to license the SH-4 core to other companies and was developing the SH-5 architecture, the first move of SuperH into the 64-bit area. The earlier SH-1 through 3 remained the property of Hitachi.

In 2003, Hitachi and Mitsubishi Electric formed a joint-venture called Renesas Technology, with Hitachi controlling 55% of it. In 2004, Renesas Technology bought STMicroelectronics's share of ownership in the SuperH Inc. and with it the licence to the SH cores. Renesas Technology later became Renesas Electronics, following their merger with NEC Electronics.

The SH-5 design supported two modes of operation: SHcompact mode, which is equivalent to the user-mode instructions of the SH-4 instruction set; and SHmedia mode, which is very different in that it uses 32-bit instructions with sixty-four 64-bit integer registers and SIMD instructions. In SHmedia mode the destination of a branch (jump) is loaded into a branch register separately from the actual branch instruction. This allows the processor to prefetch instructions for a branch without having to snoop the instruction stream. The combination of a compact 16-bit instruction encoding with a more powerful 32-bit instruction encoding is not unique to SH-5; ARM processors have a 16-bit Thumb mode (ARM licensed several patents from SuperH for Thumb ) and MIPS processors have a MIPS-16 mode. However, SH-5 differs because its backward compatibility mode is the 16-bit encoding rather than the 32-bit encoding.

The last evolutionary step happened around 2003 where the cores from SH-2 up to SH-4 were getting unified into a superscalar SH-X core which formed a kind of instruction set superset of the previous architectures, and added support for symmetric multiprocessing.

Continued availability
Since 2010, the SuperH CPU cores, architecture and products are with Renesas Electronics and the architecture is consolidated around the SH-2, SH-2A, SH-3, SH-4 and SH-4A platforms. The system-on-chip products based on SH-3, SH-4 and SH-4A microprocessors were subsequently replaced by newer generations based on licensed CPU cores from Arm Ltd., with many of the existing models still marketed and sold until March 2025 through the Renesas Product Longevity Program.

As of 2021, the SH72xx microcontrollers based on the SH-2A continue to be marketed by Renesas with guaranteed availability until February 2029, along with newer products based on several other architectures including Arm, RX, and RH850.

J Core
The last of the SH-2 patents expired in 2014. At LinuxCon Japan 2015, j-core developers presented a cleanroom reimplemention of the SH-2 ISA with extensions (known as the "J2 core" due to the unexpired trademarks). Subsequently, a design walkthrough was presented at ELC 2016.

The open source BSD-licensed VHDL code for the J2 core has been proven on Xilinx FPGAs and on ASICs manufactured on TSMC's 180 nm process, and is capable of booting μClinux. J2 is backwards ISA compatible with SH-2, implemented as a 5-stage pipeline with separate Instruction and Data memory interfaces, and a machine-generated Instruction Decoder supporting the densely packed and complex (relative to other RISC machines) ISA. Additional instructions are easy to add. J2 implements instructions for dynamic shift (using the SH-3 and later instruction patterns), extended atomic operations (used for threading primitives) and locking/interfaces for symmetric multiprocessor support. Plans to implement the SH-2A (as "J2+") and SH-4 (as "J4") instruction sets as the relevant patents expire in 2016–2017.

Several features of SuperH have been cited as motivations for designing new cores based on this architecture:

 * High code density compared to other 32-bit RISC ISAs such as ARM or MIPS important for cache and memory bandwidth performance
 * Existing compiler and operating system support (Linux, Windows Embedded, QNX )
 * Extremely low ASIC fabrication costs now that the patents are expiring (around US$0.03 for a dual-core J2 core on TSMC's 180 nm process).
 * Patent- and royalty-free (BSD-licensed) implementation
 * Full and vibrant community support
 * Availability of low cost hardware development platform for zero cost FPGA tools
 * CPU and SoC RTL generation and integration tools, producing FPGA and ASIC portable RTL and documentation
 * Clean, modern design with open source design, generation, simulation and verification environment

Models
The family of SuperH CPU cores includes:

 * SH-1 – used in microcontrollers for deeply embedded applications (CD-ROM drives, major appliances, etc.)
 * SH-2 – used in microcontrollers with higher performance requirements, also used in automotive such as engine control units or in networking applications, and also in video game consoles, like the Sega Saturn and Sega 32X add-on. The SH-2 has also found home in many automotive engine control unit applications, including Subaru, Mitsubishi, and Mazda.
 * SH-2A – The SH-2A core is an extension of the SH-2 core including a few extra instructions but most importantly moving to a superscalar architecture (it is capable of executing more than one instruction in a single cycle) and two five-stage pipelines. It also incorporates 15 register banks to facilitate an interrupt latency of 6 clock cycles. It is also strong in motor control application but also in multimedia, car audio, powertrain, automotive body control and office + building automation
 * SH-DSP – initially developed for the mobile phone market, used later in many consumer applications requiring DSP performance for JPEG compression etc.
 * SH-3 – used for mobile and handheld applications such as the Jornada, strong in Windows CE applications and market for many years in the car navigation market. The Cave CV1000, similar to the Sega NAOMI hardware's CPU, also made use of this CPU. The Korg Electribe EMX and ESX music production units also use the SH-3.
 * SH-3-DSP – used mainly in multimedia terminals and networking applications, also in printers and fax machines
 * SH-4 – used whenever high performance is required such as car multimedia terminals, video game consoles, most notably the Dreamcast, or set-top boxes
 * SH-5 – used in high-end 64-bit multimedia applications
 * SH-X – mainstream core used in various flavours (with/without DSP or FPU unit) in engine control unit, car multimedia equipment, set-top boxes or mobile phones
 * SH-Mobile – SuperH Mobile Application Processor; designed to offload application processing from the baseband LSI

SH-2

The SH-2 is a 32-bit RISC architecture with a 16-bit fixed instruction length for high code density and features a hardware multiply–accumulate (MAC) block for DSP algorithms and has a five-stage pipeline.

The SH-2 has a cache on all ROM-less devices.

It provides 16 general-purpose registers, a vector-base register, global-base register, and a procedure register.

Today the SH-2 family stretches from 32 KB of on-board flash up to ROM-less devices. It is used in a variety of different devices with differing peripherals such as CAN, Ethernet, motor-control timer unit, fast ADC and others.

SH-2A
The SH-2A is an upgrade to the SH-2 core that added some 32-bit instructions. It was announced in early 2006.

New features on the SH-2A core include:
 * Superscalar architecture: execution of 2 instructions simultaneously
 * Harvard architecture
 * Two 5-stage pipelines
 * Mixed 16-bit and 32-bit instructions
 * 15 register banks for interrupt response in 6 cycles.
 * Optional FPU

The SH-2A family today spans a wide memory field from 16 KB up to and includes many ROM-less variations. The devices feature standard peripherals such as CAN, Ethernet, USB and more as well as more application-specific peripherals such as motor control timers, TFT controllers and peripherals dedicated to automotive powertrain applications.

SH-4
The SH-4 is a RISC CPU and was developed for primary use in multimedia applications, such as Sega's Dreamcast and NAOMI game systems. It includes a much more powerful floating-point unit and additional built-in functions, along with the standard 32-bit integer processing and 16-bit instruction size.

SH-4 features include:
 * FPU with four floating-point multipliers, supporting 32-bit single-precision and 64-bit double-precision floats
 * 4D floating-point dot-product operation and matrix–vector multiplication
 * 128-bit floating-point bus allowing 3.2 GB/sec transfer rate from the data cache
 * 64-bit external data bus with 32-bit memory addressing, allowing a maximum of 4 GB addressable memory (see Byte addressing) with a transfer rate of 800 MB/sec
 * Built-in interrupt, DMA, and power management controllers

There is no FPU in the custom SH-4 made for Casio, the SH7305.

SH-5
The SH-5 is a 64-bit RISC CPU.

Almost no non-simulated SH-5 hardware was ever released, and, unlike the still-live SH-4, support for SH-5 was dropped from GCC and Linux.

title: SuperH