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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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