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We learned how to construct the map ² (V) → VV that sends the simple wedge V₁ V₂ → V₁ V₂ V₂ V₁ - Please prove that this map is injective.

We learned how to construct the map ² (V) → VV that sends the simple wedge V₁ V₂ → V₁ V₂ V₂ V₁ - Please prove that this map is injective.

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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Can you please prove if it's injective. Thank you

We learned how to construct the map A² (V) →v v that sends the simple wedge V₁ ^ V₂ → V₁ V₂ - V₂ V₁ Please prove that this map is injective. Remark: 1. we can think of the exterior product ²(V) ≤ V V instead of as a quotient. 2. In our construction, there is a natural map V V→A²(V), since it is a quotient space. Viewing the alternating maps as a subspace of the tensor product, there is no natural map back into the subspace (in other words, there isn't a natural way to make a bilinear map into an alternating map).
Transcribed Image Text:We learned how to construct the map A² (V) →v v that sends the simple wedge V₁ ^ V₂ → V₁ V₂ - V₂ V₁ Please prove that this map is injective. Remark: 1. we can think of the exterior product ²(V) ≤ V V instead of as a quotient. 2. In our construction, there is a natural map V V→A²(V), since it is a quotient space. Viewing the alternating maps as a subspace of the tensor product, there is no natural map back into the subspace (in other words, there isn't a natural way to make a bilinear map into an alternating map).
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