linear algebra proof question please show clear thanks Problem 6: Let T : V → V be a linear map and let W be a T-invariant subspace.Prove or give a counterexample to the following statements:a.) If W is a proper subspace of V, i.e., W Ç V, then any subspace of W isT-invariant.b.) If U is a subspace of V such that U W=V, then U is T-invariant.c.) Each eigenvalue A of Tw is an eigenvalue of T; moreover, the eigenspace E(X, Tw)is contained in E(X, T). (Here Tw denotes the restriction T |w: W → W of T.)

Answered: Image /qna-images/answer/a38738f6-f4dc-4a79-9dac-c37a02d577eb.jpg

Problem 6: Let T: V → V be a linear map and let W be a T-invariant subspace. Prove or give a counterexample to the following statements: a.) If W is a proper subspace of V, i.e., WV, then any subspace of W is T-invariant. b.) If U is a subspace of V such that U W = V, then U is T-invariant. c.) Each eigenvalue λ of Tw is an eigenvalue of T; moreover, the eigenspace E(X, Tw) is contained in E(X, T). (Here Tw denotes the restriction Tw: W → W of T.)

Problem 6: Let T: V → V be a linear map and let W be a T-invariant subspace. Prove or give a counterexample to the following statements: a.) If W is a proper subspace of V, i.e., WV, then any subspace of W is T-invariant. b.) If U is a subspace of V such that U W = V, then U is T-invariant. c.) Each eigenvalue λ of Tw is an eigenvalue of T; moreover, the eigenspace E(X, Tw) is contained in E(X, T). (Here Tw denotes the restriction Tw: W → W of T.)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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