I'm trying to find the matrix representation of the orthogonal projection of the subspace of $\mathbb{R^2}$ that has the basis $B=(x,-x)$. I thought it was simple enough, so I just did the usual and projected the canonical basis (where $e$ is the basis vector of the subspace):
$\langle e|e_1 \rangle e = (1,-1)$; $\langle e|e_2 \rangle e = (-1,1)$
So my thought process was that the matrix representation would just be a $2 \times 2$ matrix with those vectors as its components. But the solution in the manual includes a 1/2 factor. Can someone explain why this, or why my intuition of simply making a $2 \times 2$ matrix with those vectors is wrong?
Your projections of the standard basis vectors onto the line are wrong precisely because $e$ is not a unit vector. If you replace $e$ with the unit vector in the direction of $e$ i.e. $(1/\sqrt{2}) e$ then you get the right answer with your method.
For some more concrete intuition, notice that the squared distance from $(1,0)$ to $(x,-x)$ is $(x-1)^2+x^2=2x^2-2x+1$, which is minimized when $x=1/2$.