Define the quadratic form $Q:\mathbb{R}^{2\times 2}\to\mathbb{R}$ by $$Q(A) = \text{tr}(A^2).$$ What is the matrix representation of this bilinear form with respect to the standard basis of $\mathbb{R}^{2\times 2}$ (a $1$ in one entry and $0$ elsewhere)?
If $A = \begin{pmatrix} w\ \ x \\ y\ \ z \end{pmatrix}$ then $Q(A) = w^2+2xy+z^2$. I'm struggling to then convert this into a $2\times2$ matrix. I can find a $4\times4$ matrix to represent $Q$, namely $\begin{pmatrix} 1\ \ 0\ \ 0\ \ 0\\ 0\ \ 0\ \ 1\ \ 0\\ 0\ \ 1\ \ 0\ \ 0\\ 0\ \ 0\ \ 0\ \ 1 \end{pmatrix}$ but beyond that I'm stuck.
EDIT: My problem arose from question 3.)d.), https://i.stack.imgur.com/9RxCH.png, is this not solvable in its current form then?
