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 A = show that limn A"v = 0, where 0 is the zero vector 0,0,..., 0]. (d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an %3D 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 A = show that limn A"v = 0, where 0 is the zero vector 0,0,..., 0]. (d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an %3D 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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