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En , of V and let A : V → V be a linear map. Show that if Aei = vi = aj,i ej and bi,j = B(vi , vj ), the matrices A = [ai,j ] and B = [bi,j ] are related by: B = A+ A. (b) Show that if ν is the volume form, e ∗1 ∧ · · · ∧ e∗n , and A is orientation preserving 1 A∗ ν = (det B) 2 ν . 6 one has a bijective map Λn (V ∗ ) ∼ = An (V ) . Show that the element, Ω, of An (V ) corresponding to the form, ν, has the property |Ω(v1 , . . , vn )|2 = det([bi,j ]) where v1 , . . , vn are any n-tuple of vectors in V and bi,j = B(vi , vj ).
T + W , where W is in I k . Proof. 4, T σ = (−1)σ T + Wσ , with Wσ ∈ I k . T + W where W = (−1)σ Wσ . (−1)σ Wσ 28 Chapter 1. Multilinear algebra Corollary. I k is the kernel of Alt . Proof. We’ve already proved that if T ∈ I k , Alt (T ) = 0. 2) 1 T = − k! W. with W ∈ I k . 5. Every element, T , of L k can be written uniquely as a sum, T = T1 + T2 where T1 ∈ Ak and T2 ∈ I k . Proof. 2), T = T1 + T2 with T1 = 1 k! Alt (T ) and 1 T2 = − k! W. To prove that this decomposition is unique, suppose T 1 + T2 = 0, with T1 ∈ Ak and T2 ∈ I k .
6. Given ei ∈ V ∗ , i = 1, . . , k show that e1 ∧ · · · ∧ ek = 0 if and only if the ei ’s are linearly independent. Hint: Induction on k. 7 35 The interior product We’ll describe in this section another basic product operation on the spaces, Λk (V ∗ ). As above we’ll begin by defining this operator on the Lk (V )’s. 1) k (−1)r−1 T (v1 , . . , vr−1 , v, vr , . . , vk−1 ) ιv T (v1 , . . , vk−1 ) = r=1 on the k − 1-tuple of vectors, v1 , . . , in the r th summand on the right, v gets inserted between v r−1 and vr .
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