(b) If V = U OW is a direct sum of subspaces U, W CV then any element v E V canbe written uniquely as v = u +w with u E U and w E W. Show that T(v)defines a linear map T: V → V.= U(c) Show for the map n constructed in (b) that Im(n)U and Ker(t) = W.

Given V=U⊕W is a direct sum of subspaces U,W⊂V. Then, it is known that any element v∈V can be…

(b) If V = U OW is a direct sum of subspaces U, W C V then any element v E V can be written uniquely as v = u +w with u E U and w E W. Show that t(v) =u defines a linear map t: V → V. (c) Show for the map n constructed in (b) that Im(t) U and Ker(t) = W. %3D

(b) If V = U OW is a direct sum of subspaces U, W C V then any element v E V can be written uniquely as v = u +w with u E U and w E W. Show that t(v) =u defines a linear map t: V → V. (c) Show for the map n constructed in (b) that Im(t) U and Ker(t) = W. %3D

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.CR: Review Exercises

Problem 73CR

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Question

(b) If V = U OW is a direct sum of subspaces U, W CV then any element v E V can
be written uniquely as v = u +w with u E U and w E W. Show that T(v)
defines a linear map T: V → V.
= U
(c) Show for the map n constructed in (b) that Im(n)
U and Ker(t) = W.

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