how I can show that $T_A$ is a linear operator on $Mat_{m,n}(\mathbb{F})$:

Question

I'm trying to find out, how I can show that $T_A$ is a linear operator on $Mat_{m,n}(\mathbb{F})$:

$T_A:Mat_{m,n}(\mathbb{F}) \rightarrow Mat_{m,n}(\mathbb{F}) $

$B \rightarrow AB$

As far as I understand, I have to show that $B=AB$ and unfortunately I can't go further.

I hope someone has the time to help me a little bit.

Answer 1

What do you mean you need to show $B=AB$? This is just not true.

To show it's a linear operator, you need to show $(1)$ that $T_A(B_1+B_2)=T_A(B_1)+T_A(B_2)$ for $B_1,B_i\in M_{m,n}(F)$, and $(2)$ $T_A(\lambda B)=\lambda T_A(B)$ for $\lambda\in F$. Both of these things are just a restatement of facts about matrix multiplication that you should already know from your course.