What is the linear transformation $ x \mapsto Ax $?

Question

I am told by my textbook that,

$ \text{Nul }A = {0} \text{ if and only if the linear transformation } x \mapsto Ax \text{ is one-to-one.} $

$ \text{Col }A = \Bbb R^m \text{ if and only if the linear transformation } x \mapsto Ax \text{ maps }\Bbb R^n \text{ onto } \Bbb R^m. $

What is the linear transformation $ x \mapsto Ax $? I do not understand what this means.

Answer 1

$x \mapsto Ax$ means that they're defining a function that maps vectors $x \in \Bbb R^n$ to the vectors $(Ax)\in\Bbb R^m$ where $A$ is a $m\times n$ matrix.

There are two main ways of denoting a function:

1. The first is probably the one you're more familiar with. We can denote a function like $f: X \to Y$ given by $f(x)=y$. Often the first part of this is clear and thus left off, leaving you with just the $f(x)=y$ part. Notice that we had to name our function/ transformation here -- I named it $f$. If your problem had been written in this notation, it'd have looked like $f:\Bbb R^n \to \Bbb R^m$ given by $f(x) = Ax$.
2. We can also denote a function like $x \mapsto y$ for $x\in X$ and $y\in Y$. This is the notation that your problem chose to use. Notice that we did NOT have to name the function. So in your problem it's written like $x \mapsto Ax$ for $x\in \Bbb R^n$ and $Ax \in \Bbb R^m$.
   NOTE: You actually can name a function with this second notation. You do so like $x \stackrel{f}{\mapsto} y$. This is less common, though.

Both notations convey the same information. The $\cdot \mapsto \cdot$ notation just lets you talk about a function/ transformation without naming it. That's the only difference I've ever seen between the two.

Answer 2

well you can look at as a function ,for example $ f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with sends every $x \rightarrow x$ . Note that x is a vector here , means it has two coordinates $x_1$ and $x_2$. one way of writing this transformation is in a matrix form $x \rightarrow Ax$ with $A$ the unit matrix.(try it) you should also keep in mind that for a linear Transformation $f(ax)=af(x)$ for any a and $f(x+y)=f(x)+f(y)$