4.1.5

Let $$X, Y \in F^{m \times n}, \alpha \in F$$ and $$T(M) = AMB$$, then $$T(X + Y) = A(X + Y)B = (AX + AY)B = AXB + AYX = T(X) + T(Y)$$ and $$T(\alpha X) = A\alpha XB = \alpha AXB = \alpha T(X).$$ Therefore, $$T$$ is a linear transformation.