Question

Problem 4. Let V be a vector space and let U and W be two subspaces of V. Let (1) Prove that if U g W and W g U then UUW is not a subspace of V 2) Give an ezample of V, U and W such that U W andW ZU. Explicitly verify the implication of the statement in part (1) (3) Prove that UUW is a subspace of V if and only ifUCW or W CU.' (4) Give an example that proves the statem ent UW is a subspace of V is false. (5) Give an example of V. Ú and W such that U-W or W-Ụ. Erplicitly verify the implication of the statement in part (3). Choose a different vector space V than the one you chose in part (2) or (4). Hint: First explain the meaning of the negative sign in the RHS and the negative sign in the LHS. Lemma 1 comes in handy Hi Assume r0, show Fmust be zero. 3 I suggest that you read the math hygiene note before doing this question. Specially the sections on "connectives hegation" and contra-positive To prove "P if and only if Q" you should show "Pimplies Q" and "Q implies p separately. Usually one side is easier. In this question you already proved the contra-positive of the difficult side.

Answer 1

If U W and W U, are can find elements a and b such that a elementof U/W and b elementof W/U implies a elementof U middot U w and b elementof UUW Assume the contrans that UUW is a subspace. Then a + b elementof UUW implies that a + b elementof U or a + b elementof W if a + b elementof U then a + b - a elementof U (Since a elementof U and U is a subspace, - a elementof U) Which is contradiction b elementof W/U Thus a + b U. similarly we can prove that a + b W therefore a + b UUW Hence UUW is not a subspace (2) take V = IR^2 U = {(a, 0): a elementof R} W = {(0, b): b elementof R} U and W are Subspace of V. \r\n Also (1, 0) elementof U but not in w implies U w and (0, 1)