Let A be a square matrix of order n such that A2 = A. Prove that (a) Every v in Rn can be decomposed as v = v1 + v2, where v1 is in the nullspace of A and v2 is in the column space of A. (b) The decomposition in (i) is unique, that is, if v = v1 + v2 = v'1 + v'2 where v1, v'1 are in the nullspace of A and v2, V'2 are in the column space of A, then v1 = v'1 and v2 = V'2.

Let A be a square matrix of order n such that A2 = A. Prove that (a) Every v in Rn can be decomposed as v = v1 + v2, where v1 is in the nullspace of A and v2 is in the column space of A. (b) The decomposition in (i) is unique, that is, if v = v1 + v2 = v'1 + v'2 where v1, v'1 are in the nullspace of A and v2, V'2 are in the column space of A, then v1 = v'1 and v2 = V'2.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.6: Rank Of A Matrix And Systems Of Linear Equations

Problem 77E: Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and...

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