Linear Algebra Dimension

Question

Let $L(U,V)$ = $\{T:U\rightarrow V\ :\ T\ \text{linear}\},$ and dim $(U)=n$, dim $(V)=m$. Then show that $$ \dim L(U,V) = mn. $$ I don't know how to begin and I already searched the internet to find something, but could not. Please if anybody could help.

Answer 1

Hint. Let $e_1,\ldots,e_n$ be a basis of $U$ and $f_1,\ldots,f_m$ a basis of $V$.

Set $T_{ij}:U\to V$ to be the linear transformation such that $$ T_{ij}(c_1e_1+\cdots+c_ne_n)=c_if_j. $$ Then $T_{ij}$, $i=1,\ldots,n$, $j=1,\ldots,m$ form a basis of $L(U,V)$.