linear algebra questions

1. Let v ∈ C 4 be the vector given by v = (1, i, −1, −i). Find the matrix (with respect to the canonical basis on C 4 ) of the orthogonal projection P ∈ L(C 4 ) such that null(P) = {v} ⊥ .

2. Let U be the subspace of R 3 that coincides with the plane through the origin that is perpendicular to the vector n = (1, 1, 1) ∈ R3 . (a) Find an orthonormal basis for U. (b) Find the matrix (with respect to the canonical basis on R 3 ) of the orthogonal projection P ∈ L(R 3 ) onto U, i.e., such that range(P) = U. 

3.Let V be a finite-dimensional vector space over F with dimension n ∈ Z+, and suppose that b = (v1, v2, . . . , vn) is a basis for V . Prove that the coordinate vectors [v1]b, [v2]b, . . . , [vn]b with respect to b form a basis for F n .