Let $W$ be some inner product space over $\mathbb{R}^2$ where the inner product $\langle\,A,B\rangle=tr(AB^t)$ ($B^t$ in this case is $B$ transposed).
$w_1=\big(\begin{smallmatrix}1&1\\1&1\end{smallmatrix}\big)$, $w_2=\big(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\big)$ and also $W=span\{w1,w2\}$,
Calculate the distance of $v=\big(\begin{smallmatrix}4&0\\4&4\end{smallmatrix}\big)$ from $W$.
I know that I need to first find the projection of $v$ onto $W$, and from there it's smooth sailing. But, I don't know how to represent W. In similar questions we usually want to find the Normal of the plane, and project $v$ onto it. However in this case, I don't know how to find the normal of a span of matrices. We'd have to find a vector $U$ that is orthogonal to both $w_1,w_2$. I tried to find the orthonormal basis (using Gram–Schmidt) $B=(e_1,e_2)$, and say that $U=e_1+e_2$, but my $U$ came out to be equal to $w_2$ which didn't make much sense.
Is there something I'm missing here?