Consider the vector space V = Sym_2(R) of 2 x 2 symmetric real matrices, with the bases B = {B_1, B_ ...

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Consider the vector space V = Sym_2(R) of 2 x 2 symmetric real matrices, with the bases B = {B_1, B_2, B_3} and C = {C_1, C_2, C_3}, where B_1 = [[1, 1], [1, 0]], B_2 = [[-1, 0], [0, 1]], B_3 = [[0, -1], [-1, 1]], and C_1 = [[0, 1], [1, 0]], C_2 = [[1, 0], [0, 1]], C_3 = [[0, 0], [0, 1]]. Find the change-of-basis matrix P_{C <- B} and use it to compute the coordinate vector [v]_C, where v = [[-1, -2], [-2, 3]].