Dimension of Set of Linear Transformations Theorem

Theorem

• With $dim(V)=n<\infty$
• With $dim(W)=m<\infty$
• $dim(\mathcal{L}(V,W))=nm$

Proof

1. With $\beta =\{a_1,\dots ,a_n\}$ for $V$
2. With ${\beta }^{\prime }=\{b_1,\dots ,b_m\}$ for $W$
3. For each pair $(p,q)$ where $p\in [1,n],q\in [1,m]$
   1. Define a Linear Map $E^{p,q}\in \mathcal{L}(V,W)$

E^{p,q}(a_{i}) = \begin{cases} b_{p} & i = q \ 0 & i \neq q \end{cases}

4. Note that $\{ E^{p,q} | p \in [1,n], q \in [1,m] \}$ forms a basis for $\mathcal{L}(V,W)$ 5. Thus, $dim(\mathcal{L}(V,W)) = n * m$