Let A be a square matrix of order n such that A² = A. Prove that (a) Every v e R" can be decomposed as v = vị + v2, where vị 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 = vị + v2 = v{ +v½, where v1, ví are in the nullspace of A and v2, v', are in the column space of A, then vị = v, and v2 =

Let A be a square matrix of order n such that A² = A. Prove that (a) Every v e R" can be decomposed as v = vị + v2, where vị 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 = vị + v2 = v{ +v½, where v1, ví are in the nullspace of A and v2, v', are in the column space of A, then vị = v, and v2 =

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.6: Matrices

Problem 25E: Let A and B be square matrices of order n over Prove or disprove that the product AB is a diagonal...

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