Let A € Mmxn (C) and B ¤ Mnxp (℃). (a) Using the rank-nullity theorem, prove that rank(A) = rank(AT). (b) Prove that rank(AB) < min{rank(A), rank(B)}.
Let A € Mmxn (C) and B ¤ Mnxp (℃). (a) Using the rank-nullity theorem, prove that rank(A) = rank(AT). (b) Prove that rank(AB) < min{rank(A), rank(B)}.
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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![Let A e Mmxn (C) and Be MnXP(C).
(a) Using the rank-nullity theorem, prove that rank(A) = rank(AT).
(b) Prove that rank(AB) < min{rank(A), rank(B)}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F4141fbb7-c395-46db-a770-8a2b056e8412%2F10uy1na_processed.png&w=3840&q=75)
Transcribed Image Text:Let A e Mmxn (C) and Be MnXP(C).
(a) Using the rank-nullity theorem, prove that rank(A) = rank(AT).
(b) Prove that rank(AB) < min{rank(A), rank(B)}.
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