Let V be a vector space of dimension n over F. Recall that V V denotes...

 

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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 + U₂) v = X(U₁v) + u₂v and similarly for the second argument, i.e. u® (AV₁ + V₂) = X(u v₁) + u v₂2. Here the rules hold for all u, U₁, U₂, V, V₁, V₂ € Vand X, μ € F A decomposable 2-tensor is one of the form v w for v E 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 CVO V of symmetric 2-tensors is the span of elements of the form v v, v € V. 2. Determine the dimension of the space of symmetric 2-tensors in V V. for dim V = n. As a consequence, determine the dimension of the quotient space of "bivectors": ^²V = (VV)/(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 A²A on the vector space ^2V: We define A²A on decomposable bivectors via (^²A) (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 ^2V obtained from the choice of a basis on V. 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 + U₂) v = X(U₁v) + u₂v and similarly for the second argument, i.e. u® (AV₁ + V₂) = X(u v₁) + u v₂2. Here the rules hold for all u, U₁, U₂, V, V₁, V₂ € Vand X, μ € F A decomposable 2-tensor is one of the form v w for v E 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 CVO V of symmetric 2-tensors is the span of elements of the form v v, v € V. 2. Determine the dimension of the space of symmetric 2-tensors in V V. for dim V = n. As a consequence, determine the dimension of the quotient space of "bivectors": ^²V = (VV)/(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 A²A on the vector space ^2V: We define A²A on decomposable bivectors via (^²A) (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 ^2V obtained from the choice of a basis on V.

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1 Let dim V 2 choose a basis for V and a corresponding basis for VV and describe precisely the subset of decomposable 2tensors as a subset of F4 Let V be a vector space of dimension 2 over F with basi... View the full answer