Unitary and triangular matrices and krylov matrix

Question

Problem:

Numerical Linear Algebra : Unitary and Triangular Matrices and Krylov Matrix

1. Let A - QR be the factorization of a A into the product of a unitary matrix and a trianglular matrix. Suppose that the cloumns of A are linearly independent. Show that |r_kk| is the distance from the k-th column of A to the linear space spanned by the first k-1 columns of A.

2. Let A ∈?^mxm and b∈ ?^m abitrary. Show that any x∈K_n  is equal to p(A)b for some polynomial p of degree < n- 1.

Note: Kn is the mxn Krylov matrix