a1 - conjugate) 3. (a) Let V be the real vector space R R...

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McGill University Math 236: Algebra 2 Assignment 1: due Friday, January 20, 2006 1. (a) Show that the set R 2 , together with the operations , defined by ( x 1 ,y 1 ) ( x 2 ,y 2 ) = ( x 1 + x 2 - 1 ,y 1 + y 2 + 2) , a ( x,y ) = ( ax + 1 - a,ay - 2 + 2 a ) , is a vector space over R . (b) Show that this vector space is isomorphic to R 2 under the usual operations. Hint: Consider the mapping T defined by T ( x,y ) = ( x - 1 ,y + 2). 2. In each of the following decide whether or not W is a subspace of the vector space V over the field F = R or C . (a) V = F , W = { ( x 1 ,...,x n ,... ) V | x n +3 = x n +2 + nx n for n 1 } ; (b) V = F R , W = { f V | f ( x ) = f ( x 2 + 1) for all x R } ; (c) V = F R , W = { f V | f 00 ( x ) + xf 0 ( x ) + f ( x ) = 0 } where f 0 ( x ) is the derivative of f at x ; (d) V = F 3 , W = { ( x 1 ,x 2 ,x 3 ) R 3 | x 2 1 - x 1 x 2 + x 2 2 = 0 } ; (e) V = C 2 × 2 , W = { X V | X t = X } where X t is the transpose of X and X is the conjugate of X , the matrix obtained from X by replacing each entry by its complex
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Unformatted text preview: conjugate) 3. (a) Let V be the real vector space R R and let V even , V odd be the subsets of V consisting the even and the odd functions respectively. Show that V even , V odd are subspaces of V and that V = V even V odd . (b) Using (a), give the decomposition of the function f ( x ) = e x into its even and odd com-ponent functions. 4. If V is a vector space, prove or disprove the following statements: (a) The intersection of any family of subspaces ( W i ) i I of V is a subspace of V . (b) If U 1 ,U 2 ,W are subspaces of V with U 1 W = U 2 W then U 1 = U 2 ....
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This homework help was uploaded on 04/18/2008 for the course MATH 236 taught by Professor Toth during the Winter '06 term at McGill.

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