This question comes from the 1st edition and was asked in the Harvard lecture videos on Abstract Algebra.
Let $A$ be a $k\times m$ matrix and let $B$ be an $n\times p$ matrix. Prove that the rule $M\leadsto AMB$ defines a linear transformation from the space $F^{m\times n}$ of $m\times n$ matrices to the space $F^{k\times p}$
I am unsure of what is being asked here, as I feel it follows directly from the definitions. The domain of the rule must be $F^{m\times n}$; otherwise, the product $AMB$ is not defined. The codomain is the space of matrices with shape given by the "outer indices" of the matrices in the product. In this case, $k\times p$.
Is this answer correct? What am I missing here to make this argument formal, if it is at all valid?
You're being asked to show that the definition of linearity applies to this map. Denote $f :M \mapsto AMB$. I'll use the definition that a map $f$ is linear if for vectors $x_1,x_2$ and scalars $k_1,k_2$, we have $f(k_1x_1 + k_2x_2) = k_1f(x_1) + k_2f(x_2)$; there are several equivalent versions of this definition. For matrices $M_1,M_2$ and scalars $k_1,k_2$, we have $$ \begin{align} f(k_1M_1 + k_2M_2) &= A(k_1M_1 + k_2M_2)B \\ & = (k_1AM_1 + k_2 AM_2)B \\ & = k_1 (AM_1B) + k_2 (AM_2B) = k_1 f(M_1) + k_2 f(M_2). \end{align} $$ Thus, $f$ is a linear map.