Let T : R$^3$→R$^3$ be projection onto the plane 2x$_1$ − x$_2$ + x$_3$ = 0.
Find a basis for R$^3$ such that T is represented by a diagonal matrix in this basis.
I understand that we have to find some basis which will result in some A(x), so that A(x) is diagonal. But my confusion comes with what we need A to be. Would we be trying to find a matrix A that has the diagonals as the coefficients of the plane?
I assume that by "projection" you mean "orthogonal projection", else the question cannot be solved as stated. As $T$ is the identity on the plane (which I will denote by $\pi$), we can choose any base of $\pi\subset\mathbb{R}^3$ to obtain the first two vectors. For the third vector, we take a non-zero vector orthogonal to $\pi$. Then the eigenvalues are $1,1$ and $0$.
For example, we can take the basis given by $$\pmatrix{0\\1\\1},\pmatrix{1\\-1\\1},\pmatrix{2\\-1\\1}.$$