2. Let u e R" and v e R" be two non-zero vectors, in other words at least one component of the vectors is non-zero. Let A = uvT e R"X". (a) Suppose ||u||, = 1 and ||v||, = 1. Show that the Frobenius norm of A is equal to 1. (b) Consider the case where m = 3 and n = 2, i.e., n3= --E) --(:)- u = u2 To help simplify your work in the following subproblems you may assume u #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 mull 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 mill space of A.
2. Let u e R" and v e R" be two non-zero vectors, in other words at least one component of the vectors is non-zero. Let A = uvT e R"X". (a) Suppose ||u||, = 1 and ||v||, = 1. Show that the Frobenius norm of A is equal to 1. (b) Consider the case where m = 3 and n = 2, i.e., n3= --E) --(:)- u = u2 To help simplify your work in the following subproblems you may assume u #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 mull 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 mill space of A.
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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Transcribed Image Text:2. Let u e R" and v e R" be two non-zero vectors, in other words at least one component of the
vectors is non-zero. Let A = uvT E Rmxn.
(a) Suppose ||u||, = 1 and ||v||, = 1. Show that the Frobenius norm of A is equal to 1.
(b) Consider the case where m = 3 and n = 2, i.e.,
-E), -(:)
u =
V =
To help simplify your work in the following subproblems you may assume ui #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.
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