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2. Let F = Q, K = Q(V5), and E = K(i, 5+ i). (a) Regard K as a Q-vector space (i.e. with Q as the field of scalars). Explain why 1, V5, 25 is a basis for K. (b) Regard E as a K-vector space. Explain why {1, i} is a basis for E. (c) Regard E as an F-vector space, and find a basis for the six- dimensional vector space E.

2. Let F = Q, K = Q(V5), and E = K(i, 5+ i). (a) Regard K as a Q-vector space (i.e. with Q as the field of scalars). Explain why 1, V5, 25 is a basis for K. (b) Regard E as a K-vector space. Explain why {1, i} is a basis for E. (c) Regard E as an F-vector space, and find a basis for the six- dimensional vector space E.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CM: Cumulative Review
Problem 5CM: Take this test to review the material in Chapters 4 and 5. After you are finished, check your work...
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2. Let F = Q, K = Q(V5), and E = K(i, 5+ i). (a) Regard K as a Q-vector space (i.e. with Q as the field of scalars). Explain why 1, V5, V25 is a basis for K. (b) Regard E as a K-vector space. Explain why {1, i} is a basis for E. (c) Regard E as an F-vector space, and find a basis for the six- dimensional vector space E.
Transcribed Image Text:2. Let F = Q, K = Q(V5), and E = K(i, 5+ i). (a) Regard K as a Q-vector space (i.e. with Q as the field of scalars). Explain why 1, V5, V25 is a basis for K. (b) Regard E as a K-vector space. Explain why {1, i} is a basis for E. (c) Regard E as an F-vector space, and find a basis for the six- dimensional vector space E.
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