Basis of the space of linear maps between vector spaces

Question

I would like to know how to define a basis of the space of linear maps : $ \mathcal{L} ( E , F ) $. $ E $ and $ F $ are two differents vector spaces. I'm not looking for how building a basis of its equivalent space $ \mathcal{M}_n ( \mathbb{R} ) $, i know it. Thank you very much.

Answer 1

If we have a basis for $E$ and a basis for $F$, we can use them to produce a basis for $\mathcal{L}(E,F)$ as follows. Let $e_1,\ldots,e_n$ be a basis for $E$ and $f_1,\ldots,f_m$ a basis for $F$.

For $i\in\{1,\ldots,n\}$ and $j\in\{1,\ldots,m\}$, define the linear map $\varphi_{ij}:E\to F$ by $\varphi_{ij}\left(a_1e_1+\cdots+a_ne_n\right) = a_if_j$. Then these $mn$ linear maps $\varphi_{ij}$ form a basis for $\mathcal{L}(E,F)$.

Answer 2

Are $E$ and $F$ finite dimensional? If so, here you go.

The second paragraph here should be helpful, too.