A linear map that is multiplication by a matrix The problem statement, all given variables and data
Let $T$ be multiplication by matrix $A$:
$$A=
        \begin{bmatrix}
        1 & -1 & 3 \\
        5 & 6 & -4 \\
        7 & 4 & 2 \\
        \end{bmatrix}
$$
Find the range, kernel, rank and nullity of T.
Attempt at a solution
I can find all of these values except for A. What does the question mean when T is a multiplication by matrix A? How do I get range/kernel etc. of T.. I know how to get them for A.
Thanks.

Any matrix $A$ represents a linear function $T$, or more precisely $T_A$, defined by multiplication, that is, by $T(x)=Ax$. Now it makes sense to talk about the range and kernel of $T$. Rank and nullity are simply the dimension of range and kernel of $T$, respectively.
Regarding your question, note that, by definition or a little argument, the rank of any matrix $A$, agrees with the rank of the linear map $T_A$, and the nullity of $A$ is the same as the nullity of $T_A$.
For instance,
 $$\ker(T)=\{x\in \mathbb{R}^3\mid T(x)=\mathbf{0}\}=\{x\in \mathbb{R}^3\mid Ax=\mathbf{0}\}.$$
Thus, to find the kernel of $T$, you need to solve the homogeneous system $Ax=\mathbf{0}$.
