MTH 411-1 Abstract Algebra II S17 Homework 10 due on 4/14/17 Let K be a field and V and W be K-spaces. 1. Let f V W be a K-linear map. Put U = ker f . Show that f V /U Im f, v + U f (v) is a well-defined K-isomorphism. 2. Let X and Y be K-subspaces of V . Show that X/X Y (X + Y )/Y, x + (X Y ) x + Y is a well-defined K-isomorphism. 3. Let W be a K-subspace of V . Let S K (V, W ) be the set of K-subspaces of V containing W and let SK (V /W ) be the set of K-subspaces of V /W . Show that SK (V, W ) SK (V /W ), X X/W is a well-defined bijection. 4. Let U and W be K-subspaces of V with W U . Show that V /U K V /W /U /W. 5. Let f V W be K-linear function. If V is finite dimensional, Show that Im f is finite dimensional and dim V = dim Kerf + dim Im f Hint: Use Exercise 1 and the Dimension Formula. 6. Let X and Y be finite dimensional K-subspaces of V . Show that X + Y is finite dimensional and dimK (X + Y ) = dimK X + dimK Y dimK (X Y ) Hint: Use Exercise 2 and the Dimension Formula. 7. Let K be a field and (Ki )iI a family of subfields of K. Put E = Ki . iI Show that E is a subfield of K. 8. Let F K be a field extension and a K. Put F(a) = {xy 1 x, y F[a], y 0K }. Show that F(a) is subfield of K and F(a) = {E E is a subfield of K, F E, a E}. 1