Unsure about notation with matrix

Question

I'm not sure about a notation:

Let $X= A\mathbb R^3$ where $A$ is a $3$x$3$ matrix. What is $X$? I think it should be the image of $\mathbb R^3$ under the transformation with matrix $A$ but I am not sure.

Answer 1

You've got it right. It's the image of the transformation given by $A$. (You have to specify a basis first for this to be meaningful!)

Alternately, with the standard basis assumed, $$ X = A \Bbb{R}^3 = \left\{ A \vec{v} \mid \vec{v} \in \Bbb{R}^3 \right\} = \operatorname{Col} A $$ is the column space of the matrix $A$. It is a linear subspace of $\Bbb{R}^3$ spanned by the columns of $A$.