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%3D 14. (a) Suppose W1 and W2 are subspaces in a vector space V and that V W1 + W2. Prove that V W,eW, if and only if any vector v E V can be represented uniquely as v = v + vz where vy E W1, t2 E W2. (b) Suppose W and W2 are subspaces in a vector space V, and such that V = W1W2. If B, is a basis of W, and B, is a basis of W, prove that B, UB, is a basis of V. (c) If W, and W, are invariant subspaces for a linear transformation T: V V and V = W, eW2 prove that A1 0 A2 (a block matrix) where A, Tw, and Az-Ziv,

%3D 14. (a) Suppose W1 and W2 are subspaces in a vector space V and that V W1 + W2. Prove that V W,eW, if and only if any vector v E V can be represented uniquely as v = v + vz where vy E W1, t2 E W2. (b) Suppose W and W2 are subspaces in a vector space V, and such that V = W1W2. If B, is a basis of W, and B, is a basis of W, prove that B, UB, is a basis of V. (c) If W, and W, are invariant subspaces for a linear transformation T: V V and V = W, eW2 prove that A1 0 A2 (a block matrix) where A, Tw, and Az-Ziv,

Elementary Linear Algebra (MindTap Course List)
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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14. (a) Suppose W, and W are subspaces in a vector space V and that V = W + W2. Prove that V = W,eW, if and only if any vector vEV can be represented uniquely %3D as v = v + v2 where vy E Wi, 2 E W2. (b) Suppose W, and W2 are subspaces in a vector space V, and such that V= W,eW2. If 3, is a basis of W, and B, is a basis of W2 prove that B, UB, is a basis of V. (c) If W and W, are invariant subspaces for a linear transformation T: V V and V = W, eW2 prove that A1 A (a block matrix) where A, 1Tw, and A-Zv,K
Transcribed Image Text:14. (a) Suppose W, and W are subspaces in a vector space V and that V = W + W2. Prove that V = W,eW, if and only if any vector vEV can be represented uniquely %3D as v = v + v2 where vy E Wi, 2 E W2. (b) Suppose W, and W2 are subspaces in a vector space V, and such that V= W,eW2. If 3, is a basis of W, and B, is a basis of W2 prove that B, UB, is a basis of V. (c) If W and W, are invariant subspaces for a linear transformation T: V V and V = W, eW2 prove that A1 A (a block matrix) where A, 1Tw, and A-Zv,K
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