1. Let A be an m x n matrix. Show that:(a) If x € N (AT A), then Ax e R(A) N N (AT).(b) N (A" A) = N(A).(c) A and A"A have the same rank.(d) If A has linearly independent columns, then AT A is nonsingular.

Let A be an m×n matrix. Show that: (a) If x∈NATA, then Ax∈RA∩NAT (b) NATA=NA. (c) A and AT have…

Answered: Let A be an m x n matrix. Show that: (…

Let A be an m x n matrix. Show that: ( a ) Ifxε Ν (Α'Α), then AxE R(Α) Ω Ν (A"). (b) N (AT A) = N(A). (c) A and AT A have the same rank.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.8: Determinants

Problem 34E

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Question

1. Let A be an m x n matrix. Show that:
(a) If x € N (AT A), then Ax e R(A) N N (AT).
(b) N (A" A) = N(A).
(c) A and A"A have the same rank.
(d) If A has linearly independent columns, then AT A is nonsingular.

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