Prove that ϕ⊗ψ is a R-bilinear form on V × V

1. Quiz 11

Let V be a finite dimensional vector space over R. A linear functional on V is a linear map ϕ : V → R. The set of all linear functionals on V forms a vector space over R denoted by V^∗.

(1) Let ϕ, ψ ∈ V^∗. The tensor product of ϕ and ψ is a function ϕ ⊗ ψ : V × V = V² → R

defined by (ϕ⊗ψ)(v, w) = ϕ(v)ψ(w) for (v, w) ∈ V². Prove that ϕ⊗ψ is a R-bilinear form on V × V.

(2) The set of all bilinear forms on V is denoted by T²(V ). Prove that T²(V ) forms a real vector space. (We call T²(V ) the space of 2-tensors on V.)

(3) Let {ϕ₁, · · · , ϕ_n} be a basis for V (in other words, we assume that dim V = n). Prove that {ϕi⊗ ϕ_j : 1 ≤ i < j ≤ n} forms a basis for T²(V ). Therefore dim T²(V ) = n². (4) Let ϕ, ψ ∈ V^∗. We define the wedge product of ϕ and ψ by

(ϕ ∧ ψ) = ϕ ⊗ ψ − ψ ⊗ ϕ ∈ T²(V ).

Prove that ϕ ∧ ψ : V × V → R is a skew-symmetric bilinear form and that (ϕ ∧ ψ)(v, w) =

ϕ(v) ϕ(w) ψ(v) ψ(w)     . Also prove that ψ ∧ ψ = 0 for any ψ ∈ V^∗.

(5) The space of skew symmetric bilinear forms on V is denoted by Λ²(V ). Show that Λ²(V ) forms a vector subspace of T²(V ).

(6) Let {ϕ₁, · · · , ϕ_n} be as above a basis for V^∗. Show that {ϕ_i∧ ϕ_j : 1 ≤ i, j ≤ n}

forms a basis for Λ²(V ). Compute dim Λ²(V ).

(7) Let p be a nonnegative integer. Let ψ1, · · · , ψp ∈ V^∗. We define ψ1∧ ψ₂∧ · · · ∧ ψ_p: V^p → R by

(ψ₁∧ ψ₂∧ · · · ∧ ψ_p)(v₁, · · · , v_p) = det(ψ_i(v_j))_1≤i,j≤p.

Prove that ψ₁∧ ψ₂∧ · · · ∧ ψ_p is alternating p-linear form on V. (Use the property of determinants.)

Remark. A p-linear form f : V^p → R is said to be alternating if for any permutation σ on {1, · · · , p} (a permutation on {1, · · · , p} is a bijection), we have

f (vσ1, · · · , vσp) = (sgn σ)f (v1, · · · , vp).

Here sgn σ = 1 if σ is an even permutation and sgn σ = −1 when σ is an odd permutation.

(8) The space of all alternating p-linear forms on V forms a vector space over R and is denoted by Λ^p(V ).

(9) Prove that Λ^p(V ) = 0 when p > n.

(10) Prove that ΛⁿV is spanned by a single vector ϕ1∧ · · · ∧ ϕ_n; hence dim ΛⁿV = 1.

(11) Prove that {ϕ_i1 ∧ · · · ∧ ϕ_ip : 1 ≤ i₁ < i₂ < · · · < i_p ≤ n} forms a basis for Λ^p(V ) when 1 ≤ p ≤ n Compute dim Λ^p(V ).

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