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.
Basis of the space of linear maps between vector spaces
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linear-algebra
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0Sorry, I'm a Morrocan men, I don't speak well english. – 2012-12-13
2 Answers
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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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Are $E$ and $F$ finite dimensional? If so, here you go.
The second paragraph here should be helpful, too.
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0Yes, $ E $ and $ F $ are finie dimensional. Thanks :) – 2012-12-14