Let v1 = (0, 3, -1) , v2 = (1, 2, 0) , and let W be the plane spanned by v1 and v2. Consider the function T : R 3 → R 3 where T(x) = projW x , that is, T(x) is the projection of x onto the plane W; you may assume that T is a linear transformation.
i) Evaluate T(5, 0, 10).
ii) Find a basis for W⊥ in R 3.
iii) Without calculation, write down the matrix of T with respect to the ordered basis { v1, v2, v3 }, where v3 is the basis element found in part (ii). By drawing a diagram, or otherwise, give reasons for your answer.
iv) Hence or otherwise, find an expression for the matrix of T with respect to the standard basis in R 3. You may leave your answer as a product of matrices without completing the calculation.