1. Let A ∈ Fm×n, with F = R or C, prove that range(A) is orthogonal to null(A∗), i.e., any arbitrary vector in range(A) is orthogonal to an arbitrary vector in null(A∗).
2. Use Gram-Schmidt method to find a QR factorization of the matrix
Ñ1 2 1é
A = 3 −1 1 .
1 1 2
3. Consider a matrix A ∈ Fm×n with m ≥ n and all columns being orthogonal but not of unit length, what should its reduced QR decomposition look like?
