2. Let u e Rm and v E R" be two non-zero vectors, in other words at least one component of the vectors is non-zero. Let A = uv' e R™xn. (a) Suppose ||u||2 = 1 and ||v|l2 = 1. Show that the Frobenius norm of A is equal to 1. (b) Consider the case where m = 3 and n = 2, i.e.., u = U2 v = V2 U3 To help simplify your work in the following subproblems you may assume u1 # 0 and vi # 0. i. Derive a basis for the range of A using Gaussian elimination. What is the rank of A? ii. Derive a basis for the null space of A using Gaussian elimination. (c) Now consider the general case where m and n are any positive integers. To help simplify your work in the following subproblems you may assume u1 #0 and vi #0. i. Generalize your work from b.i to derive a basis for the range of A. What is the rank of A? ii. Generalize your work from b.ii to derive a basis for the null space of A.
2. Let u e Rm and v E R" be two non-zero vectors, in other words at least one component of the vectors is non-zero. Let A = uv' e R™xn. (a) Suppose ||u||2 = 1 and ||v|l2 = 1. Show that the Frobenius norm of A is equal to 1. (b) Consider the case where m = 3 and n = 2, i.e.., u = U2 v = V2 U3 To help simplify your work in the following subproblems you may assume u1 # 0 and vi # 0. i. Derive a basis for the range of A using Gaussian elimination. What is the rank of A? ii. Derive a basis for the null space of A using Gaussian elimination. (c) Now consider the general case where m and n are any positive integers. To help simplify your work in the following subproblems you may assume u1 #0 and vi #0. i. Generalize your work from b.i to derive a basis for the range of A. What is the rank of A? ii. Generalize your work from b.ii to derive a basis for the null space of A.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.2: Inner Product Spaces
Problem 101E: Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that...
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