Given $4$ spanning $2 \times 2$ matrices, how would you find a basis? Our professor will not let us isomorph them to $\mathbb R^4$. I tried putting them into elementary matrices however I do not know what to do afterwards.
2026-03-25 22:03:36.1774476216
How would you go about finding a basis of matrices in a vector space?
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Hint: Can you show that the set $$ \bigg\{\begin{pmatrix} 1&0\\ 0&0 \end{pmatrix}, \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}, \begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}, \begin{pmatrix} 0&0\\ 0&1 \end{pmatrix}\bigg\}$$ is linearly independent? If so, then it must be spanning because $\dim(\mathcal{M}_2(\mathbf{R}))=4$.