1 Mark as TRUE or FALSE. If they are true give an explanation, if they are false, give a coun- terexample. You really need to understand the reasons behind your statements. (a) Every square matrix has an eigenvalue over the real numbers. (b) Let A be an n x n matrix. A vector v can be an eigenvector with respect to two different eigenvalues. (c) If the vector v is an eingenvector for the eigenvalue = show that limn-0 A"v = 0, where 0 is the zero vector (0, 0, ..., O]. (d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an eigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c-1. (e) 5 vectors in R are always linearly dependent. (f) 4 vectors in R are always linearly independent.
1 Mark as TRUE or FALSE. If they are true give an explanation, if they are false, give a coun- terexample. You really need to understand the reasons behind your statements. (a) Every square matrix has an eigenvalue over the real numbers. (b) Let A be an n x n matrix. A vector v can be an eigenvector with respect to two different eigenvalues. (c) If the vector v is an eingenvector for the eigenvalue = show that limn-0 A"v = 0, where 0 is the zero vector (0, 0, ..., O]. (d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an eigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c-1. (e) 5 vectors in R are always linearly dependent. (f) 4 vectors in R are always linearly independent.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 35EQ
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