The vector normal to W is . Given a vector in , the effect of F on will be to add a multiple of to so as to take it to the subspace W. The condition for to belong to W is . Solve that to see that . From that, you should be able to write down the matrix A and verify that .

For the last part, if A is the matrix of the orthogonal projection onto a subspace, then I–A is the matrix of the orthogonal projection onto the (orthogonal) complementary subspace.