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 (Au + ) v = X(U v) + Uv and similarly for the second argument, i.e. u (Av + ) = X(u v) + uv. Here the rules hold for all u, U, U2, V, V, V Vand , 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 SV 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)/(SV). 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 A A on the vector space AV: We define A 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