Talk:Linear Algebra/Linear Transformations

I added a concrete example of a nonlinear transformation S early on:

"For example, the transformation S (whose input and output are both vetors in R2) defined by

$$S\mathbf{x} = S\begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix}  xy\\ cos(y)\end{pmatrix}$$

is non-linear (for example, it does not fix the origin). Linear transformations are easy to study, and any transformation is nearly linear if one looks at it closely enough, i.e. zooms in on it. Hence one can shed light on complicated nonlinear transformations by studying easier, linear ones."

hopefully this is a good idea? i found that when i first learned about linear transformations i had little motivation as to why we study them. do apologise if i'm messing someone's book up! rmijic

Some TODOs
This page needs some notes on null space/column space and rank/nullity. (Nullity is already briefly mentioned.)

The Point
The thing that makes this article so important is that it describes in relatively readable terms why matrices exist and what matrix multiply is for. Prior to reading this I had wondered for years why anyone would invent something as convoluted as matrix math, despite having taken a course in linear algebra. One of the major problems with math articles in wiki* and elsewhere is that the articles are almost always written on a level far higher than the subject matter. They are written by people familiar with the material for people who have already learned it and just need a refresher (undoubtedly this is usually unintentional). Thus it is a very steep learning curve for those trying to learn new material.

My point is simply this: If you are going to edit or add onto this sort of article, please remember to make your additions couched in the simplest terms you can without diluting the material. As Einstein said: "Make everything as simple as possible, but not simpler." Anuran 15:44, 1 April 2006 (UTC)

preserves identity
"preserves identity: T0 = 0" .... um surely this is preserving the null vector? non-trivial linear transformations do not preserve the identity, this might be confusing? Sorry, too far out of my field to correct personally.

Isomorphism
It has been written that $$T$$ is an isomorphism if it is bijective and an inverse exists (among other conditions).

But if $$T$$ is bijective, then its inverse does exist, doesn't it?