Talk:Linear map

Examples question

the example matrix of "rotation by 90 degrees counterclockwise" under "Examples of linear transformation matrices" -- doesnt this example rotate 90 degrees clockwise? or is it meant to work on the computer-screen type of coordinates, where +y axis is down?

Thanks for any clarification, i'm just starting to learn about this stuff. This is a great resource. — Preceding unsigned comment added by 67.160.123.125 (talk) 27:21, 10 March 2007 (UTC)

Terminology wrong!

I have known linear algebra for decades, and have seen many books on the subject. Every book has used "transformation" as the main term, with "map" used informally later (presumably because the author gets too lazy to write "transformation"). According to google, "linear map" is more common than "linear transformation", but again, I am inclined to attribute this to laziness or the existence of other uses of "linear map" (i.e., linear function specified by a linear polynomial). I move that the main term is changed to "transformation", as well as the name of the article, but maybe my bias is cultural.

More importantly, the statement that a "transformation" is often from a set to itself is totally bogus. The literature is clear that "transformation" is synonymous with "function" or "map", but "operator" has a tendency to mean a function with inputs and outputs drawn from the same set. Not only does my own experience say this, but so does the American Heritage Dictionary[1]. Can anyone back me up here? I think we really need to change this in the introduction, and I also posted something to the "operator" wikipedia page, which I think is also factually incorrect. —Preceding unsigned comment added by B2smith (talk • contribs) 23:26, 9 December 2008 (UTC)

The terminology is not wrong, but correct. Which 2 terms you prefer is a matter of personal taste, but either is ok. I don't think a a lengthy (and most likely rather subjective) discussion, which term might be slightly "more common" or "more appropriate" is of any use for the article.--Kmhkmh (talk) 00:40, 2 March 2009 (UTC)

The first commenter is basically correct in both points. The usual term is "linear transformation". That is the most commonly recognized term. (Google is certainly no authority.) There are indeed some people who prefer "map" and it is used, so it's not "wrong". Nevertheless the title should be changed back on the ground of greatest familiarity. (I also teach linear algebra.) Zaslav (talk) 07:32, 26 February 2011 (UTC)

"Linear transformation" is unnecessarily long, old fashioned and misleading: it is sometimes used to mean that the domain and codomain are the same, sometimes not. If you do not believe me, search for "linear transformation of" in Google books. You will also find lots of informal usages such as "linear transformation of a random variable", "linear transformation of the Lebesgue integral on R". Many of the textbooks are old with usages such as "linear transformation of V into W" which may suggest injectivity when it is not intended. In contrast, many modern textbooks, as well as some modern classics, use "linear map". See:

• A. Knapp, Basic Algebra
• K. Janich, Linear Algebra
• H. E. Rose, Linear Algebra, a pure mathematical approach
• S. Lang, Introduction to linear algebra
• S. Axler, Linear algebra done right

A good mathematician is a lazy mathematician: far from being informal, the term "linear map" is precise: it is a map which is linear, nothing more, nothing less. Geometry guy 14:23, 26 February 2011 (UTC)

It's hard not to think Geometry guy's last remarks are pretty silly (as well as POV). Unless he really is The Ultimate Authority. Geometry guy, please explain what makes your opinions authoritative. This is not a snide remark, I really want to know. My credentials: I have been a working mathematician for several decades. Notice that I am not claiming that makes me an Authority. It means I have a lot of experience. Zaslav (talk) 03:29, 27 February 2011 (UTC)

I am the only editor commenting here (so far) with the ability to refer to reliable secondary sources. Geometry guy 03:40, 27 February 2011 (UTC)

There are plenty of examples of books using either term, see these Google Books searches:

• "linear map"
• "linear transformation"

However It might be instructive to read some of the books which mention both terms:

• both

Paul August ☎ 13:49, 27 February 2011 (UTC)

Differentiating the vectors from numbers of the field K

Why don't we use ${\vec {x}}$ or $\mathbf {x}$ instead of $x$ to denote vectors?--Netheril96 (talk) 01:47, 15 October 2010 (UTC)

Usually lower-case Latin letters are vectors, lowercase greek letters are scalars, and upper-case Latin letters are vector spaces. Thus, no confusion. Bender2k14 (talk) 23:05, 31 January 2011 (UTC)

Bender's right that there's no confusion, but I think saying that it is the "usual" convention is a bit introverted. It is the convention usually used by pure mathematicians, whereas bold Latin letters for vectors is the one usually used by applied mathematicians, physicists, engineers, etc. By sheer weight of numbers, I imagine the latter convention is much more common. The real answer to the question is that this article is a pure mathematics one, so that convention makes most sense here. Of course either convention is clear so long as it is used consistently. Quietbritishjim (talk) 17:03, 26 February 2011 (UTC)

Change of Basis Section

The section about change-of-basis matrices is written very badly. There are multiple grammatical errors. His explanation barely makes sense to me, and I already understand what he's trying to say. I could go into further detail about what needs to be fixed, but I'm hoping that others will see what I mean. Omarschall (talk) 02:27, 19 December 2011 (UTC)

Definition and first consequences

The claim "For example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear" needs a proof. The claim should be removed if it's not proved right there. — Preceding unsigned comment added by ThinkerFeeler (talk • contribs) 06:58, 23 January 2012 (UTC)

Proof that f(0) = 0

I'm not a regular user of Wikipedia, so I won't change this (I can't figure out how to do it without messing up the math notation). But in my view, the current proof that the linearity of f implies that f(0) = 0 leaves a bit to be desired.

The last equality of this proof asserts without comment that the result of scaling the vector f(0) by the scalar 0 is the 0 vector. The issue is that "0v = v for all vectors v" is not one of the vector space axioms. Rather, it is a quick but nontrivial consequence of these axioms. And its proof from (the axioms) is of roughly "the same difficulty" as a proof (from the axioms and the definition of linearity) that f(0) = 0. So the current proof that f(0) = 0, although not wrong, is certainly less "elementary" and "self-contained" than it might otherwise be.

Here is a chain of equalities establishing that f(0) = 0 without appealing to nontrivial consequences of the vector space axioms.

0 = f(0) + (-f(0)) = f(0 + 0) + (-f(0)) = (f(0) + f(0)) + (-f(0)) = f(0) + (f(0) + -f(0)) = f(0) + 0 = f(0).

Note that each equality in this chain is an immediate specialization of either a single vector space axiom or the additivity of f (and note that the current proof does not have this property). FWIW, this chain also shows that only the additivity of f is needed to deduce that f(0) = 0; homogeneity is not required.

IMVHO the rest of the page's content would benefit greatly from being re-organized somewhat to distinguish more carefully between fundamentals (common to all linear transformations) and specific examples. At the moment, there is a prominent and untamed zoo of 2x2 matrix examples this page, and quite a lot of discussion of theory that recapitulates stuff that seems (to me) to be more appropriate for pages on more specific topics (e.g. the discussion of cokernel, and "algebraic classification of linear transformations") 68.40.167.85 (talk) —Preceding undated comment added 00:31, 1 June 2012 (UTC)

"a linear map is a homomorphism of modules" – incorrect?

A homomorphism of modules does not imply homogeneity: one can have a homomorphism of modules in which the respective scalar rings are merely isomorphic, but not even the same ring; alternatively where they are the same ring but the homorphism involves a non-identity automorphism between the scalar rings of the two modules. This statement would therefore have to be modified to read something like "a linear map is a homomorphism of modules over the same ring". — Quondum 07:40, 15 December 2012 (UTC)

Change of basis

It seems to me that much in this section is improper and confusing. However, I am not very familiar with covariance and contravariance, so I will not try to fix this myself. I would prefer that

vectors be boldface;

the matrix $A$ and basis $B$ use different notation styles;

the inverse transformation not be represented by the basis $B$;

the inverse transformation not use vectors $v$ and $v'$.

Where do $u'$ and $A'$ come from? — Anita5192 (talk) 16:54, 14 April 2016 (UTC)

Proposition: Latex Conversion

Proposition. Latex should be preferred default for mathematical rendering on this page because:

1. Usage of multiple templates or raw unicode leads to a LOT of rendering variability and browser interaction, whereas Latex focuses efforts of mathematical typography and rendering onto preferred community libraries. This makes usability very hard to test.

2. Latex provides many fallback options, including rendering to SVG or PNG.

3. It's easier for amateur Wikipedians to copy-paste Latex, and it's easier to follow along when the community has consistent style.

4. Maintenance becomes easier with uniformity. Across different math pages one may find raw unicode, a no-wrap template, a variables template, or a generic Math template.

5. Editing by source becomes very ugly with multiple styles.

6. Mathematical typography should be consistent at least within-page even if not between pages.

7. More popular peer math pages prefer this style, such as Linear Algebra.

SirMeowMeow (talk) 14:18, 22 January 2021 (UTC)

Linear extensions

Mgkrupa has added a definition of a "linear extension" in section § Definition and first consequences. As this terminology seemed uncommon in the context of this article, I tagged it with {{citation needed}}. Then, he added a reference to a book in operator theory. This and a quick Google-Scholar search suggest that the term is used only with topological vector spaces, and that a linear extension must be continuous.

So, the definition added by Mgkrupa differs from that which is commonly used, and is out of context here (no topology is involved). This is for these reasons that I have removed this definition. For being added again, a reliable source must be provided for attesting that the term is commonly used in a purely algebraic context. D.Lazard (talk) 08:46, 9 February 2022 (UTC)

Your removal of sourced relevant content is disruptive editing. Cite the Wikipedia policy to justify its removal. Your "no topology is involved" argument is not a valid reason. Even if linear extensions only appeared in the context of TVSs, so what? Mgkrupa 14:57, 9 February 2022 (UTC)

so what? In that case, I'd intuitively vote for removal of the sentence here. "Linear map" is about a very general concept of which TVS theory is one application, and we certainly wouldn't introduce all definitions made in all applications. If "linear extension" is important in linear algebra, some (advanced) linear algebra textbook should mention it. - Jochen Burghardt (talk) 15:43, 9 February 2022 (UTC)

First, here is just one linear algebra book that uses the term "linear extension". Second, I'd like to address your "Linear map" is about a very general concept... and (advanced) linear algebra textbook statements about generality and advanced linear algebra material. It's important to keep in mind this article's primary target audience, which will be students learning elementary linear algebra. (See Wikipedia:Manual of Style/Mathematics, and Wikipedia:Make technical articles understandable, and Wikipedia:Manual of Style for more details.) Consideration of the target audience is why − just like the articles Matrix (mathematics) and Vector space − this article's main focus is not on the most general possible concept of a linear map; linear maps − in their greatest generality (e.g. extension of scalars, etc.) − can be discussed at the end of this article, which is exactly what the articles Matrix (mathematics) and Vector space do. Finally, I'd like to point out the "if" in my statement "Even if ..., so what?" Mgkrupa 16:49, 9 February 2022 (UTC)

To be clear, this is the content that D.Lazard removed:

A linear extension of a function $f$ refers to an extension of $f$ to some larger vector space that is also a linear map.

The following is one example of how this concept is used in linear algebra: if ${\mathcal {B}}$ is a given set of linearly independent vectors and $f:{\mathcal {B}}\to Y$ is a function, then $f$ has a linear extension to $\operatorname {span} {\mathcal {B}}.$ Here is a more concrete example that is representative of how this is used: "Extend $(1,0)\to 1$ and $(0,1)\to 2$ to a linear map $F:\mathbb {R} ^{2}\to \mathbb {R} .$" Mgkrupa 17:35, 9 February 2022 (UTC) With regard to D.Lazard's concerns about topologies, note that no topology was involved. Mgkrupa 20:28, 9 February 2022 (UTC)

Also, I'd like to point out that linear extensions plays a fundamental role in many other common statements/terminology that appear in linear algebra and related fields. For example, it is an essential part of each of the following phrases: (a) extend by linearity, (b) extend linearly, (c) extend this-or-that to a linear map/functional/transformation, etc. ... So all of these terms are relevant to this discussion and should also be considered when investigating how often this concept appears. Mgkrupa 18:44, 9 February 2022 (UTC)

Anyone have any more comments or concerns? Mgkrupa 19:59, 15 February 2022 (UTC)

(For some reason, your comments of 9 Feb didn't cause the page to appear in my watchlist.) My in that case referred to your if. Since your now provided a linear-algebra textbook as a source, I'd be fine with your sentence. (I assume Smith gives your definition; I couldn't find that in the book excerpts Google allows me to read.)

However, I wonder if f itself must have a vector space as its domain, as your sentence suggests; this doesn't apply to your ${\mathcal {B}}$ example? Similarly, f itself can hardly be linear on ${\mathcal {B}}$ , while the "also" in your sentence suggests it is. - Jochen Burghardt (talk) 20:34, 15 February 2022 (UTC)

"the "also" in your sentence suggests it is" When I wrote the "also" in "also a linear map", I intended for it to indicate that the extension was a linear map; I did not consider the possibility that some readers could instead interpret the "also" as an indication that "$f$ is linear on ${\mathcal {B}}$". My apologies for the confusion - the word "also" is not needed and can/should be removed. $f$ does not need to be defined on a vector space. Mgkrupa 19:36, 16 February 2022 (UTC)

I misread both "also" (as "like f itself") and "larger" (as "comparison between vector spaces, not sets"); now I understand your intended meaning. In the article, I deleted both words; this shouldn't change the meaning, since "extension" already implies the superset relation on the involved domain sets. - As another question: Why don't you cite Smith in the article? - Jochen Burghardt (talk) 07:59, 17 February 2022 (UTC)

You mean this book? There are plenty of other sources that I could use besides him. And I already have citations. Why would I spend the time to add another? Mgkrupa 04:03, 19 February 2022 (UTC)

Yes, I meant this one. I'd prefer it since it is obviously a linear-algebra textbook, while the plentytwo cited sources are about operator theory and analysis. - Jochen Burghardt (talk) 16:18, 19 February 2022 (UTC)

References