Talk:Linear map

message: 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)
thread_title: Examples question

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

text: 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)
thread_title:  Differentiating the vectors from numbers of the field K 

message: 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)
thread_title:  Change of Basis Section 

message: 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)
thread_title:  Definition and first consequences 

message: 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)
thread_title:  Proof that f(0) = 0 

message: 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)
thread_title:  "a linear map is a homomorphism of modules" – incorrect? 

message: 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)
thread_title:  Change of basis 

message: 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)
thread_title:  Proposition: Latex Conversion