Moderate# Orthogonal Complement of a Matrix Subspace

LINALG-BG1D4F

Let $V=M_2(\mathbb{R})$ the vector space of $2x2$ real matrices with respect to matrix addition and scalar multiplication. This vector space can be endowed with an inner product defined as follows:

$$\langle A,B\rangle=tr(AB^{T})$$

...where $A,B\in V$ and $tr()$ means the trace of the matrix and $B^{T}$ is the transpose of $B$.

For $W=Span\biggl(\begin{pmatrix} 1 & 1\\\ 0&1 \end{pmatrix}\biggr)$ determine $dim(W^{\perp})$.