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November 13th, 2016, 09:57 PM  #1 
Newbie Joined: Nov 2016 From: United Kingdom Posts: 4 Thanks: 0  Invertible basis elements! Help!
Does there exist a basis for M2×2(R) in which each basis element is invertible? Justify your answer. 
November 14th, 2016, 04:20 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,456 Thanks: 1337 
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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basis, elements, invertible 
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