Exercise 2.6.9

If $T$ is linear, we can set $f_i$ be $g_{i}T$ as the Hint. Since it’s composition of two linear function, it’s linear. So we have

For the converse, let ${\{e_i\}}_{i=1,2,\dots ,m}$ be the standard basis of ${𝔽}^m$. So if we have that $T(x)={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^m{f}_i(x)e_i$ with $f_i$ linear, we can define $T_i(x)=f_i(x)e_i$ and it would be a linear transformation in $L({𝔽}^n,{𝔽}^m)$. Thus we know $T$ is linear since $T$ is summation of all $T_i$.