<s>[INST] Find a basis for $W$ and identify $\dim(W)$. Let $\mathbf{R}^{3\times3}$ be the linear space of all $3 \times 3$ matrices. 
Let $W$ be the set of all symmetric $3\times3$ matrices. Then
$W$ is a linear subspace of $\mathbf{R}^{3×3}$. Find a basis for $W$ and identify $\dim(W)$.
Would the $\dim(W)$ be $3$ since it is a $3\times3$ matrix. Is that correct?  How do you go about finding the basis? From my understanding, if the span of $W$ are linearly independent then they would be the basis of the $3\times3$ matrix.
Please help! 
:) Thanks
 [/INST] The $3\times3$ symmetric matrices are those of the form$$\begin{pmatrix}a&b&c\\b&d&e\\c&e&f\end{pmatrix}.$$They form a $6$-dimensional space. A basis of this space is$$\left\{\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix},\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix},\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix},\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix},\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix},\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}\right\}.$$Can you prove it?
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