18. A vector space V is said to be the direct sum of its subspaces U and W, written V = U®W, if every vector in V can be expressed in exactly one way as v = u+ w, where u is a vector in U and w is a vector in W. a. Prove that V = U@W if and only if every vector in V is the sum of some vector in U and some vector in W and UnW = {0}. %3D b. Let U be the xy-plane and W the z-axis in R'. Is it true that R = U©W? Explain. c. Let U be the xy-plane and W the yz-plane in R'. Can every vector in R' be expressed as the sum of a vector in U and a vector in W? Is it true that R' = UOW? Explain. %3D

18. A vector space V is said to be the direct sum of its subspaces U and W, written V = U®W, if every vector in V can be expressed in exactly one way as v = u+ w, where u is a vector in U and w is a vector in W. a. Prove that V = U@W if and only if every vector in V is the sum of some vector in U and some vector in W and UnW = {0}. %3D b. Let U be the xy-plane and W the z-axis in R'. Is it true that R = U©W? Explain. c. Let U be the xy-plane and W the yz-plane in R'. Can every vector in R' be expressed as the sum of a vector in U and a vector in W? Is it true that R' = UOW? Explain. %3D

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...

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