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Prove the following statement. The set B = (v₁,,Un) is a basis of an F-vector space V if and only if each vector w in V can be written in a unique way as a combination W = V₁x₁ + +v+nxn = BX.

Prove the following statement. The set B = (v₁,,Un) is a basis of an F-vector space V if and only if each vector w in V can be written in a unique way as a combination W = V₁x₁ + +v+nxn = BX.

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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Prove the following statement. The set B (V₁,,Un) is a basis of an F-vector space V if and only if each vector w in V can be written in a unique way as a combination W = V₁x₁ + +v+nxn = BX. Prove the following statement. Let S = (v₁,, Un) be a subset of a vector space V, and let : F→ V be the map defined by (X) = SX. Then (i) (ii) (iii) is injective if and only if S is independent, is surjective if and only if S spans V, and is bijective if and only if S is a basis of V.
Transcribed Image Text:Prove the following statement. The set B (V₁,,Un) is a basis of an F-vector space V if and only if each vector w in V can be written in a unique way as a combination W = V₁x₁ + +v+nxn = BX. Prove the following statement. Let S = (v₁,, Un) be a subset of a vector space V, and let : F→ V be the map defined by (X) = SX. Then (i) (ii) (iii) is injective if and only if S is independent, is surjective if and only if S spans V, and is bijective if and only if S is a basis of V.
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