Q4 (10 points) Let V be a vector space of dimension n over F. Recall that...

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Q4 (10 points) Let V be a vector space of dimension n over F. Recall that V V denotes the vector space (also over F) of 2-tensors, i.e. formal linear combinations of bilinear expressions of the form u v for u, v € V. In the above, bilinear means that we assume the formal rules (Au₁ + ₂) v = X(U₁ v) + U₂v and similarly for the second argument, i.e. u (Av₁ + ₂) = X(u v₁) + uv₂. Here the rules hold for all u, U₁, U2, V, V₁, V₂ € Vand λ, μ € F. A decomposable 2-tensor is one of the form v w for v € V. 1. Let dim V = 2, choose a basis for V and a corresponding basis for VV, and describe precisely the subset of decomposable 2-tensors as a subset of F4 The subspace S²V CVV of symmetric 2-tensors is the span of elements of the form vv, v E V. 2. Determine the dimension of the space of symmetric 2-tensors in VV, for dim V = n. As a consequence, determine the dimension of the quotient space of "bivectors". V = (V ® V)/(S²V). Let : VO V → A2V be the quotient map. We define the wedge product of v, w V to be the image of vw under this quotient map: v/w = n(vw) 3. Using a basis for V, build a basis for ^2V using the above wedge product notation. 4. Let v, w€ V. Prove that (v, w) is linearly independent if and only if v w ‡ 0. 5. Let A: V → V be an operator. Then A defines a linear operator A² A on the vector space A²V: We define A² A on decomposable bivectors via ^^ (v ^ w) = (Av) ^ (Aw), and to define it on all of A2V we simply require that it must be a linear map. Let V have dimension 2 and let A be an operator on V. Write the matrix of A2 A relative to a basis for A2V obtained from the choice of a basis on V Q4 (10 points) Let V be a vector space of dimension n over F. Recall that V V denotes the vector space (also over F) of 2-tensors, i.e. formal linear combinations of bilinear expressions of the form u v for u, v € V. In the above, bilinear means that we assume the formal rules (Au₁ + ₂) v = X(U₁ v) + U₂v and similarly for the second argument, i.e. u (Av₁ + ₂) = X(u v₁) + uv₂. Here the rules hold for all u, U₁, U2, V, V₁, V₂ € Vand λ, μ € F. A decomposable 2-tensor is one of the form v w for v € V. 1. Let dim V = 2, choose a basis for V and a corresponding basis for VV, and describe precisely the subset of decomposable 2-tensors as a subset of F4 The subspace S²V CVV of symmetric 2-tensors is the span of elements of the form vv, v E V. 2. Determine the dimension of the space of symmetric 2-tensors in VV, for dim V = n. As a consequence, determine the dimension of the quotient space of "bivectors". V = (V ® V)/(S²V). Let : VO V → A2V be the quotient map. We define the wedge product of v, w V to be the image of vw under this quotient map: v/w = n(vw) 3. Using a basis for V, build a basis for ^2V using the above wedge product notation. 4. Let v, w€ V. Prove that (v, w) is linearly independent if and only if v w ‡ 0. 5. Let A: V → V be an operator. Then A defines a linear operator A² A on the vector space A²V: We define A² A on decomposable bivectors via ^^ (v ^ w) = (Av) ^ (Aw), and to define it on all of A2V we simply require that it must be a linear map. Let V have dimension 2 and let A be an operator on V. Write the matrix of A2 A relative to a basis for A2V obtained from the choice of a basis on V