Math Woods | Free Mathematics Problems and Concepts

Solution

Solution by Anduril · EN

$\mathcal{B} = \{ I_n \} \cup \{ I_n + E_{ii} \mid 2 \le i \le n \} \cup \{ I_n + E_{ij} \mid 1 \le i, j \le n, i \neq j \}$ works
In finite dimension, subspaces are closed. Hence, $\operatorname{Vect}(GL_n(\mathbb{R})) = \overline{\operatorname{Vect}(GL_n(\mathbb{R}))}$ which contains $\overline{GL_{n}(\mathbb{R})}$ (closure preserves inclusion and a set is included in the space it spans) which is $M_{n}(\mathbb{R})$ . So finally $Vect(Gl_{n}(\mathbb{R}))=M_{n}(\mathbb{R})$ ie $M_{n}(\mathbb{R})$ has a basis of invertible matrices because, according to the basis extraction theorem, a generative set contains a basis.