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Prove that the set of $2\times 2$ matrices $H = \{A\in M_2(\mathbb{F}) | A[1,1] = [0,0]\}$ is a subspace of $M_2(\mathbb{F})$ and find a basis for $H$.

EDIT: I can prove it's a subspace by applying the subspace test. H contains the zero vector, clearly. Then H is closed under addition because (A+B)[1,1] = A[1,1] + B[1,1] = [0,0]. Finally scalar multiplication is simple to prove. Not sure how to find a basis however.

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"Hint" : Let $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, then $A\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}\Leftrightarrow a=-b\wedge c=-d$

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