(b) For subspaces X and Y, the sum X + Y is a direct sum (denoted X + Y) if every VEX + Y can be written uniquely as v = x + y, where x € X and y € Y. Prove that X + Y is a direct sum if and only if X ^ Y = {0}.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.5: Subspaces, Basis, Dimension, And Rank
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(b) For subspaces X and Y, the sum X + Y is a direct sum (denoted XY) if every
ve X + Y can be written uniquely as v = x + y, where x € X and y € Y. Prove that
X + Y is a direct sum if and only if X ^ Y = {0}.
Transcribed Image Text:(b) For subspaces X and Y, the sum X + Y is a direct sum (denoted XY) if every ve X + Y can be written uniquely as v = x + y, where x € X and y € Y. Prove that X + Y is a direct sum if and only if X ^ Y = {0}.
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