Let V the space vector of 2x2 matrices in IR. Let W be the subspace of symmetric matrices

Question

I studied this subject a time ago, but now I have to answer this question. But the big problem is that my old exercises are different. I can't find a similar question on internet. Can you help me?

-- Let V the vector space of 2x2 matrices in $\mathbb{R}^{2 \times 2}$. Let W be the subspace of symmetric matrices. Show that dim W = 3, determining a basis of W.

Can you help me? Should I use kernel?

Answer 1

Hint:

Rather than thinking of $2\times 2$ matrices: $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ which are symmetric, you can think of the related question of vectors $\begin{bmatrix}a\\b\\c\\d\end{bmatrix}$ if that helps you think.

Additional hint:

A symmetric $2\times 2$ matrix of the form $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ must have $b=c$

Answer 2

Let $B_1=\begin{bmatrix}1&0\\0&0\end{bmatrix}, B_2=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ and $B_3=\begin{bmatrix}0&0\\0&1\end{bmatrix}$.

Then each $B_j$ is symmetric. Show that $\{B_1,B_2,B_3\}$ is linearly independent and that $W=span (\{B_1,B_2,B_3\})$.