The space of all linear maps from $U\to V$ is isomorphic to $\text{Mat}_{l\times k}(K).$

Question

While trying to solve the following problem

Let $K$ be a finite field of $q$ elements. Let $U$, $V$ be vector spaces over $K$ with $\dim(U) = k$, $\dim(V) = l$. How many linear maps $U \rightarrow V$ are there?

I came across the following claim here:

The space of all linear maps from $U\to V$ is isomorphic to $\text{Mat}_{l\times k}(K).$

Could someone link me to a proof of this result?

score 2 · Accepted Answer · answered Jun 16 '19 at 06:56

2

It's solved here: Proving isomorphism between linear maps and matrices

And the proof is based on Matrix representation of linear transformations

answered Jun 16 '19 at 06:56

Raúl Astete

441

The space of all linear maps from $U\to V$ is isomorphic to $\text{Mat}_{l\times k}(K).$

1 Answers1