Problem 2. Let T be a given linear transformation. 1. Explain in detail why every non-zero scalar multiple of an eigenvector of T is also an eigenvector of T. 2. Show that every non-zero linear combination of two eigenvectors v, w corresponding to the same eigenvalue is also an eigenvector. 3. Prove that a linear combination cv + dw (with c, d 0) of two eigenvectors corresponding to different eigenvalues is never an eigenvector. Problem 3. In this problem we investigate the relationship between the eigenvalues of a matrix and its determinant. 1. We will take for granted that = det(A) A1 A2 Ak, that is, the determinant of a matrix is equal to the product of its eigenvalues. Prove that this is true if A is diagonalizable. 2. Prove that if | det(A)| > 1, then A has at least one eigenvalue with |λ| > 1. Hint: Try arguing by contradiction. 3. If | det(A)| < 1, are all the eigenvalues |X| < 1? Prove or find a counter-example.

Problem 2. Let T be a given linear transformation. 1. Explain in detail why every non-zero scalar multiple of an eigenvector of T is also an eigenvector of T. 2. Show that every non-zero linear combination of two eigenvectors v, w corresponding to the same eigenvalue is also an eigenvector. 3. Prove that a linear combination cv + dw (with c, d 0) of two eigenvectors corresponding to different eigenvalues is never an eigenvector. Problem 3. In this problem we investigate the relationship between the eigenvalues of a matrix and its determinant. 1. We will take for granted that = det(A) A1 A2 Ak, that is, the determinant of a matrix is equal to the product of its eigenvalues. Prove that this is true if A is diagonalizable. 2. Prove that if | det(A)| > 1, then A has at least one eigenvalue with |λ| > 1. Hint: Try arguing by contradiction. 3. If | det(A)| < 1, are all the eigenvalues |X| < 1? Prove or find a counter-example.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 1BEXP

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