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5. Let V be a vector space (note that this is an arbitrary vector space - not necessarily F"). Suppose dim(V) = n. Prove that a set of vectors {v₁,..., Vn} in V is linearly independent if and only if it spans V. Hint 1: Remember (from sec. 1.6) that there is always an isomorphism A: V → F and isomorphisms "preserve" bases (and so preserves linear independence). Hint 2: Think in terms of pivots.

5. Let V be a vector space (note that this is an arbitrary vector space - not necessarily F"). Suppose dim(V) = n. Prove that a set of vectors {v₁,..., Vn} in V is linearly independent if and only if it spans V. Hint 1: Remember (from sec. 1.6) that there is always an isomorphism A: V → F and isomorphisms "preserve" bases (and so preserves linear independence). Hint 2: Think in terms of pivots.

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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5. Let V be a vector space (note that this is an arbitrary vector space - not necessarily F"). Suppose dim(V) = n. Prove that a set of vectors (v₁,..., Vn} in V is linearly independent if and only if it spans V. • Hint 1: Remember (from sec. 1.6) that there is always an isomorphism A: V → F and isomorphisms "preserve" bases (and so preserves linear independence). Hint 2: Think in terms of pivots.
Transcribed Image Text:5. Let V be a vector space (note that this is an arbitrary vector space - not necessarily F"). Suppose dim(V) = n. Prove that a set of vectors (v₁,..., Vn} in V is linearly independent if and only if it spans V. • Hint 1: Remember (from sec. 1.6) that there is always an isomorphism A: V → F and isomorphisms "preserve" bases (and so preserves linear independence). Hint 2: Think in terms of pivots.
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