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 |rkk| 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∈Kn is equal to p(A)b for some polynomial p of degree < n- 1.
Note: Kn is the mxn Krylov matrix