MATH 465/565: Grassmannian Notes
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چکیده
Example. Every non-zero element of ⋀ V is trivially totally decomposable. Example. If dimV = 3, then every non-zero element of ⋀ V is totally decomposable. Proof. Given a sum v1 ∧v2 +v3 ∧v4 of two non-zero elements of ⋀ V , the two-dimensional subspaces Span(v1, v2) and Span(v3, v4) must intersect. If w1 is in the intersection, then we can rewrite v1∧v2 = w1 ∧w2 for some w1 ∈ Span(v1, v2). Similarly we can rewrite v3 ∧ v4 = w1 ∧w3 for some w3. Then v1 ∧ v2 + v3 ∧ v4 = w1 ∧w2 +w1 ∧w3 = w1 ∧ (w2 +w3)
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