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4. Let V be a vector space and let L : V → V be a linear transformation. Let A be a scalar. (a) Define V, the A-eigenspace, and prove that it is a subspace of V. (b) Let v, and v, be eigenvectors with eigenval- ues A1 and A2, respectively. If A1 7 12, prove that v, and V2 are linearly independent.

4. Let V be a vector space and let L : V → V be a linear transformation. Let A be a scalar. (a) Define V, the A-eigenspace, and prove that it is a subspace of V. (b) Let v, and v, be eigenvectors with eigenval- ues A1 and A2, respectively. If A1 7 12, prove that v, and V2 are linearly independent.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 11CM
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4. Let V be a vector space and let L : V → V be a linear transformation. Let A be a scalar. (a) Define V, the X-eigenspace, and prove that it is a subspace of V. (b) Let be eigenvectors with eigenval- ues A1 and A2, respectively. If A1 # 12, prove that v, and v, are linearly independent. V1 and V2 V2 are linearly independent.
Transcribed Image Text:4. Let V be a vector space and let L : V → V be a linear transformation. Let A be a scalar. (a) Define V, the X-eigenspace, and prove that it is a subspace of V. (b) Let be eigenvectors with eigenval- ues A1 and A2, respectively. If A1 # 12, prove that v, and v, are linearly independent. V1 and V2 V2 are linearly independent.
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