11. Let A be an m xn matrix and let S be the set of vectors consisting of the rows of A. (a) Use the Simplified Span Method to show that dim(span(S)) =rank(A). (b) Use the Independence Test Method to prove that dim(span(S)) =rank(AT). (c) Use parts (a) and (b) to prove that rank(A) = rank(A").

11. Let A be an m xn matrix and let S be the set of vectors consisting of the rows of A. (a) Use the Simplified Span Method to show that dim(span(S)) =rank(A). (b) Use the Independence Test Method to prove that dim(span(S)) =rank(AT). (c) Use parts (a) and (b) to prove that rank(A) = rank(A").

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.2: Norms And Distance Functions

Problem 33EQ

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Question

11. Let A be an m xn matrix and let S be the set of vectors consisting of the
rows of A.
(a) Use the Simplified Span Method to show that
dim(span(S)) =rank(A).
(b) Use the Independence Test Method to prove that
dim(span(S)) =rank(AT).
(c) Use parts (a) and (b) to prove that rank(A) = rank(A").

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