Size of Set of Linear Transformations Theorem

Theorem

• With $dim(V)=n$
• $dim(W)=m$
• Then, $n,m<\infty$
• Then, $dim(L(V,W))=mn$

Proof

• Let $\beta =\{{\alpha }_1,\dots ,{\alpha }_n\}$ be an ordered basis
• Let ${\beta }^{{}^{\prime }}=\{{\beta }_1,\dots ,{\beta }_m\}$ be an ordered basis
• For $V,W$, for each pair $(p,q)$ with $1\le p\le m,1\le q\le n$
• Define a map $E_{pq}\in L(V,W)$ with :

$$E_{pq}(a_i)=\{\begin{array}{l}{B}_p,i=q\\ 0,i\ne q\end{array}={\delta }_{ip}{\beta }_p,{\delta }_{ip}=\{\begin{array}{l}1,i=q\\ 0,i\ne q\end{array}$$

We can show that if $T\in L(V,W)$, then $T={\sum }_n^{i=1}{\sum }_m^{j=1}{E}_{ij}$ and $\{E_{ij}|1\le i\le m,1\le j\le m\}$ is Linearly Independent