Construct an isomorphism from the vector space of symmetric 2 x 2 matrices onto R^3
Let $\displaystyle L:M_{22}\rightarrow\mathbb{R}^3$ s.t. $\displaystyle L\!\left(\begin{bmatrix}a & b \\ b & c\end{bmatrix}\right)=\begin{bmatrix}a \\ b \\ c\end{bmatrix}$ where $\displaystyle a,b,c\in\mathbb{R}$.
I leave it for you to show that this is indeed an isomorphism.