Decomposing space of 2x2 matrices over the reals

[Mathkid3242]

Homework Statement

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.

Homework Equations

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.[/B]

 

[member 587159]

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