# Basis of the space of linear maps between vector spaces

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.

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Please, try to make the title of your question more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. – Julian Kuelshammer Dec 13 '12 at 15:32
Sorry, I'm a Morrocan men, I don't speak well english. – Bryan Dec 13 '12 at 15:37

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 $\mathcal{L}(E,F)$.

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Thank you very much – Bryan Dec 14 '12 at 19:18

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

The second paragraph here should be helpful, too.

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Yes, $E$ and $F$ are finie dimensional. Thanks :) – Bryan Dec 14 '12 at 19:19