math110s-hw4

math110s-hw4 - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 4 If V is a vector space over F , we will write dim F ( V ) for the dimension of V when we wish to emphasize the field of scalars. For example, dim C ( C 3 ) = 3 and dim R ( C 3 ) = 6. 1. Let W 1 , W 2 be subspaces of V such that V = W 1 W 2 . Let W be a subspace of V . Show that if W 1 W or W 2 W , then W = ( W W 1 ) ( W W 2 ) . Is this still true if we omit the condition ‘ W 1 W or W 2 W ’? 2. For the following vector spaces V , find the coordinate representation of the respective elements. (a) V = P 2 = { ax 2 + bx + c | a,b,c R } . Find [ p ( x )] B where p ( x ) = 2 x 2 - 5 x + 6 , B = [1 ,x - 1 , ( x - 1) 2 ] . (b) V = R 2 × 2 . Find [ A ] B where A = ± 2 3 4 - 7 ² , B = ±± 1 1 1 1 ² , ± 0 - 1 1 0 ² , ± 1 - 1 0 0 ² , ± 1 0 0 0 ²² . (c) V = R 2 . Let θ R be fixed. Find [ v ] B where v = ± x y ² , B = ±± cos θ sin θ ² , ± - sin θ cos θ ²² .
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This note was uploaded on 08/01/2008 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at Berkeley.

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math110s-hw4 - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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