"Let $V$ be a finite-dimensional vector space and T be the projection on $W$ along $W'$, where $W$ and $W'$ are subspaces of $V$. Find an ordered basis $\beta$ for $V$ such that $[T]_\beta$ is a diagonal matrix."
Playing around in $R^2$ and $R^3$ I found it difficult to reach a diagonal matrix. E.g. let $\beta = \{(1 1 1),(101) \}$ be a basis of $W$ and $\gamma = \{ (001)\}$ be a basis for $W'$. Then $T(abc) = (aba)$. Any basis of $V$ must contain some vector $v_i$ with a value at $a$, e.g. $(100)$ which means that $T(100) = (101)$, which again means that in one column of $[T]_\beta $there will be more than one value, so that it can't be a diagonal matrix.
Now... what did I get wrong?
thanks !