By Meng-Gen, Tsai

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Since W/T (W ) is finite-dimensional. By part (b), dim W + T (V )/T (V ) = dim W/W ∩ T (V ). By part (c), dim V /W + T (V ) = dim W ∩ T (V )/T (W ). So V /W + T (V ) and W + T (V )/T (V ) are finite-dimensional, that is, V /T (V ) is finite-dimensional. Again we apply Problem 2-49(c) and thus we obtain dim V /T (V ) = dim V /W + T (V ) + dim W + T (V )/T (V ) Therefore, dim V /T (V ) = dim W/T (W ). Problem 2-54. What does M being free on m1 , · · · , mn say in terms of the elements of M? Solution.

A/b) ∈ MP (V ). So ϕ(MQ (W )) ⊆ MP (V ). Problem 2-22*. Let T : An → An be an affine change of coordinates, T (P ) = Q. Show that T : OQ (An ) → OP (An ) is an isomorphism. Show that T induces an isomorphism from OQ (V T ) to OP (V ) if P ∈ V . 32 Proof. Since T is an affine change of coordinates, T −1 exists. Thus, by Problem 2-21, T : OQ (An ) → OP (An ), T −1 : OP (An ) → OQ (An ) are well-defined. By Problem 2-6, T ◦ T −1 = T −1 ◦ T = 1 = identity on OP (An ), T −1 ◦ T = T ◦ T −1 = 1 = identity on OQ (An ).

If Y = 0, then X = 0 or (X, Y ) = (0, 0). If Y 3 = −1, then 0 = Y (Y 2 +X) = Y 3 +XY = −1+XY or XY = 1; 0 = X(X 2 +Y ) = X 3 +XY = X 3 +1 or X 3 = −1. In this case, G = X 3 + Y 3 + 3XY = (−1) + (−1) + 3 = 1 = 0, that is, the points satisfying X 3 = Y 3 = −XY = −1 are not in G. So (0, 0) is the multiple point of G. Write G = 3XY + higher terms. So the tangent lines at (0, 0) are X and Y . Therefore, (1, 1) is the multiple point of F , and the tangent lines at (1, 1) are X − 1 and Y − 1. (d) Let F = Y 2 + (X 2 − 5)(4X 4 − 20X 2 + 25) = Y 2 + (X 2 − 5)(2X 2 − 5)2 .

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