Question

If A is m × n and B is n × m, show that AB = 0 if and only if col B ? null A. Let A be an m × n matrix with columns c1, c2, ..., cn. If rank A = n, show that {ATc1, ATc2, ..., ATcn} is a basis of ?n.

          If A is m × n and B is n × m, show that AB = 0 if and only if col B ? null A.
Let A be an m × n matrix with columns c1, c2, ..., cn. If rank A = n, show that {ATc1, ATc2, ..., ATcn} is a basis of ?n.

Added by Lindsey P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Krishna G

03/19/2022

Step 1

(=>) Suppose AB = 0. Then for any column vector b in B, we have Ab = 0, which means b is in the null space of A. Show more…

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If A is m x n and B is n x m, show that AB = 0 if and only if col B ⊆ null A. Let A be an m x n matrix with columns c1, c2, ..., cn. If rank A = n, show that {ATc1, ATc2, ..., ATcn} is a basis of Rn.