5. Let A be an n x k matrix for some 1 ≤ k ≤n, with real entries. 5a. Show that nul(ATA) = nul(A). 5b. Show that rank(ATA) = rank(A). 5c. Show that AT A is invertible if and only if A has independent columns.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
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5. Let A be an n x k matrix for some 1 ≤ k ≤n, with real entries.
5a. Show that nul(AT A) = nul(A).
5b. Show that rank(ATA) = rank(A).
5c. Show that AT A is invertible if and only if A has independent columns.
Transcribed Image Text:5. Let A be an n x k matrix for some 1 ≤ k ≤n, with real entries. 5a. Show that nul(AT A) = nul(A). 5b. Show that rank(ATA) = rank(A). 5c. Show that AT A is invertible if and only if A has independent columns.
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