# Isomorphic?

Algebra Level pending

Let $$\mathbb{E}, \mathbb{F}$$ be vectorial spaces with finite dimension over a field $$\mathbb{K}$$ and $$\mathbb{V}$$ a subspace of $$\mathbb{E}$$. In these conditions, let's denote $$\mathscr{M}$$ the vectorial subspace of $$\mathcal{L} \left(\mathbb{E}, \mathbb{F} \right)$$ whose elements are linear applications from $$\mathbb{E}$$ to $$\mathbb{F}$$ such that for every vector of $$\mathbb{V}$$ its image is $$0_{\mathbb{F}}$$. Let $$\mathbb{W}$$ be a vectorial subspace of $$\mathbb{E}$$ such that $$\mathbb{E}=\mathbb{V} \oplus \mathbb{W}$$.

Then, $$\mathscr{M}$$ and $$\mathcal{L} \left(\mathbb{W}, \mathbb{F} \right)$$ are isomorphic spaces.

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