Let V = span((1, 1, 0), (0, 0, 1)). Let pr v : R 3 R 3 denote the linear transformation whose value on a vector w R 3 is equal to pr v (w), the...

Let V = span((1, 1, 0), (0, 0, 1)). Let prv : R3 → R3 denote the linear transformation whose value on a vector w ∈ R3 is equal to prv (w), the orthogonal projection of w onto V . Use the Dimension Theorem (Theorem 12.1 from the notes) to prove that the nullity of prv is equal to 1. And then find the standard matrix A of prv : R3 → R3.

Theorem 12.1 (Dimension Theorem). Let F : Rn → Rm be a linear transformation. Then 

rank of F + nullity of F = n.