Invariance properties of transformations

Question

In Gentle's Matrix Algebra (2007, p. 175), he presents a table of what features of vectors various transformations preserve.

What does it mean to say a transformation T preserves some property of a vector v (if that is indeed what is meant)?

Are there some simple examples to illustrate how this concept works, and how I can verify to myself that the claims in the table are true?

Answer 1

Linear maps don't necessarily preserve individual vectors (but when they do, those vectors are called eigenvectors with eigenvalue $1$). This really comes down to how you define line, colinear, angle, and length. Here's what I would use:

1. A line is a set of the form $\{u+rv:r\in\mathbb{R}\}$ for some $u,v\in V$ with $v\ne 0$.
2. Three points in $V$ are colinear if there is a line that contains those points.
3. The angle between $u,v\in V$ is usually defined as the quantity $$\arccos\left(\frac{\langle x,y \rangle}{|x||y|}\right).$$ Here $V$ must be an inner product space (e.g. $\mathbb{R}^n$ with the dot product).
4. The length of $v\in V$ is just the norm of $V$. If $V$ is an inner product space, the length of $v$ is $\langle v, v \rangle^{1/2}$.

For example, let $f:V\to W$ be a linear map and let $L=\{u+rv:r\in\mathbb{R}\}$ be a line. If we assume that $f(v)\ne 0$, then $f(L)=\{f(u)+rf(v):r\in\mathbb{R}\}$, which is also a line. (What is the geometric interpretation in the case $f(v)=0$?)

Another example: if $f:V\to V$ is a scaling map, then there is some $r>0$ such that $f(x)=rx$ for all $x\in V$. If $u,v\in V$, then the angle between $f(u)$ and $f(v)$ is $$ \arccos\left(\frac{\langle rx,ry \rangle}{|rx||ry|}\right) = \arccos\left(\frac{\langle x,y \rangle}{|x||y|}\right),$$ which is the same as the angle between $u$ and $v$. (Can you see why the $r$'s cancel?)