Let v isin c4 be the vector given by v 1 i -1 -i find the

Question

1. Let v ∈ C4 be the vector given by v = (1, i, -1, -i). Find the matrix (with respect to the canonical basis on C4) of the orthogonal projection P ∈ L(C4) such that null(P ) = {v}⊥ .
2. Let U be the subspace of R3 that coincides with the plane through the origin that is perpendicular to the vector n = (1, 1, 1) ∈ R3.
   1. (a)  Find an orthonormal basis for U.
   2. (b)  Find the matrix (with respect to the canonical basis on R3) of the orthogonal projection P ∈ L(R3) 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 Fn.