How can a matrix act on a set

Question

Question from a PhD entrance exam

If $A=$ \begin{bmatrix} 2&-1\\-1&2 \end{bmatrix}

and $X=\{x\in \Bbb R^2:\|x\|<1\}$ where $\|x\|=|x_1|+|x_2|$

Find $AX$.

Now I know that how $\|x\|<1$ looks like.It looks like https://i.stack.imgur.com/g09Pd.png

But I dont know what is meant by $AX$ .

How can a matrix act on a set?

Can someone please help me.

EDIT

Using @RobertIsrael

The extreme points are $(1,0),(0,1),(0,-1),(-1,0)$

Now by multiplying $A$ with each one of them we get $A(1,0)=(2,-1)^T,A(0,1)=(-1,2)^T,A(-1,0)=(-2,1)^T,A(0,-1)=(1,-2)^T$

Answer 1

The notation $AX$ is almost certainly shorthand for the set $$ \{Ax\ |\ x\in X\} $$ which is the image of the set $X$ under the map $A$.

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Here are a couple of ideas to get you started on characterizing $AX$. I recommend you work them out on your own --- especially if you plan to be taking a PhD entrance exam any time soon!

• Draw a picture of the action of $A$ on $\mathbb{R}^2$. In conjunction with your picture of $X$, this will help you understand what $AX$ will be.
• Characterize $X$ more precisely than simply drawing a picture. One way of doing it is to write elements of $X$ as linear combinations of some set, i.e., specify $v_1,v_2$ such that $X = \{c_1v_1 + c_2v_2\ |\ c_1,c_2\mbox{ satisfy some constraint}\}$.
• Use the fact that $A$ is a linear map to draw conclusions about $AX$ from your characterization of $X$.