# Decomposing space of 2x2 matrices over the reals

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1. Mar 31, 2017

### Mathkid3242

1. The problem statement, all variables and given/known data
Consider the subspace $$W:=\Bigl \{ \begin{bmatrix} a & b \\ b & a \end{bmatrix} : a,b \in \mathbb{R}\Bigr \}$$ of $$\mathbb{M}^2(\mathbb{R}).$$

I have a few questions about how this can be decomposed.

1) Is there a subspace $$V$$ of $$\mathbb{M}^2(\mathbb{R})$$ such that $$W\oplus V=\mathbb{M}^2(\mathbb{R})$$? If so, what is one?

2) Further, is there a different (i.e., $$\ne V$$) subspace with the same property? And if not, is there a different proper subspace $$U$$ such that $$W+U= \mathbb{M}^2(\mathbb{R})$$?
Which examples, if any, would work for these questions? I haven't made much progress, so seeing explicit examples would help.

2. Relevant equations

3. The attempt at a solution

I tried considering this. Set $$V= \{A\in \mathbb{M}^2(\mathbb{R}): A^T =-A\}$$, but it didn't work.

2. Mar 31, 2017

### Math_QED

A good start would be to find the dimension of $W$. Then, you know that the dimension of $V$, if it exists, should be $4 - dim(W)$.

Last edited: Mar 31, 2017