Concern an m x n matrix A with a reduced singular value decomposition, A = U_r DV_r^T, and the pseudoinverse A⁺ = V_r D^–1U^T_r.

Verify the properties of A⁺:

a. For each y in R^m, AA⁺y is the orthogonal projection of y onto Col A.

b. For each x in Rⁿ, A⁺Ax is the orthogonal projection of x onto Row A.

c. AA⁺A = A and A⁺ AA⁺ = A⁺.