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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 &#1028; C^(mxm) and b &#1028; C^m be abitrary. Show that anyy x &#1028; Kn is equal to p(A)b for some polynomial p of degree < n- 1.
Note: Kn is the mxn Krylov matrix

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Unitary and Triangular Matrices and Krylov Matrix are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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