Linear Transformation Characterized by Finite Number of Scalars

Let $V$, $W$ be finite dimensional vector spaces with $dim(V)=n$ and $dim(W)=k$

Any linear transformation $T:V\to W$ is uniquely identified by $nk$ scalars.

Intuition

Fix bases $\alpha =\{v_1,\dots ,v_n\}$ $\beta =\{w_1,\dots ,w_n\}$ With $V=span(\alpha )$ $W=span(\beta )$

Proof

We know that $T$ is uniquely determined by their actions on a basis. Thus, we have $T(v_i)\subset W$ $⟹T(v_1)=a_{11}\overrightarrow{w_1}+..+a_{1k}\overrightarrow{w_k}$ $⋮$ $T(v_n)=a_{n1}\overrightarrow{w_1}+\cdots +a_{nk}\overrightarrow{w_k}$ then, there are $nk=dim(V)\ast dim(W)$ amount of coefficients with determine the map $T:V\to W$.