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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