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November 13th, 2016, 09:57 PM   #1
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Invertible basis elements! Help!

Does there exist a basis for M2×2(R) in which each basis element is invertible?
Justify your answer.
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November 14th, 2016, 04:20 PM   #2
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the 4 matrices

$\begin{pmatrix}1 &0 \\ 0 &1 \end{pmatrix}$

$\begin{pmatrix}0 &1 \\ 1 &0 \end{pmatrix}$

$\begin{pmatrix}1 &0 \\ 0 &-1 \end{pmatrix}$

$\begin{pmatrix}0 &-1 \\ 1 &0 \end{pmatrix}$

form a basis for $\mathbb{R}_{2\times2}$ and are each invertible.

I leave it to you to show they are linearly independent and span $\mathbb{R}_{2\times2}$. It's not hard.
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