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(a) Construct the natural (independent of basis) linear map from V to V". Prove that the linear map is an isomorphism. (b) If u₁,..., um are linearly independent elements of V', prove that dim(null(u₁) null(u₂)...null(um)) = dimV — m - (Hint: Double dual basis)

(a) Construct the natural (independent of basis) linear map from V to V". Prove that the linear map is an isomorphism. (b) If u₁,..., um are linearly independent elements of V', prove that dim(null(u₁) null(u₂)...null(um)) = dimV — m - (Hint: Double dual basis)

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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(a) Construct the natural (independent of basis) linear map from V to V". Prove that the linear map is an isomorphism. (b) If u₁,..., Um are linearly independent elements of V', prove that dim(null(u₁) null(u₂) Ñ ... Ñ null(um)) = dimV - m (Hint: Double dual basis)
Transcribed Image Text:(a) Construct the natural (independent of basis) linear map from V to V". Prove that the linear map is an isomorphism. (b) If u₁,..., Um are linearly independent elements of V', prove that dim(null(u₁) null(u₂) Ñ ... Ñ null(um)) = dimV - m (Hint: Double dual basis)
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