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Unformatted text preview: Math 235 Sample Midterm  1 Answers NOTE :  Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Give the definition of an inner product h , i on a vector space V . Solution: h , i : V V R such that h ~v,~v i > 0 for all ~v 6 = ~ 0 and h ~ , ~ i = 0. h ~v, ~w i = h ~w,~v i h ~v,a~w + b~x i = a h ~v, ~w i + b h ~v,~x i b) Let B = { ~v 1 ,...,~v n } be orthonormal in an inner product space V and let ~v V such that ~v = a 1 ~v 1 + + a n ~v n . Prove that a i = < ~v,~v i > . Solution: Taking the inner product of both sides with ~v i to get < ~v,~v i > = < a 1 ~v 1 + + a n ~v n ,~v i > = a 1 < ~v 1 ,~v i > + + a n < ~v n ,~v i > = a i since B is orthonormal. c) Define what it means for a set B to be orthonormal in an inner product space V . Solution: A set B = { ~v 1 ,...,~v n } is orthonormal in V if < ~v i ,~v j > = 0 for all i 6 = j and < ~v i ,~v i > = 1 for 1 i n . d) State the RankNullity Theorem. Solution: Suppose that V is an ndimensional vector space and that L : V W is a linear mapping into a vector space W . Then rank( L ) + nullity( L ) = dim V = n e) Find the rank and nullity of the linear mapping L : R 3 M 2 2 ( R ) defined by L ( x 1 ,x 2 ,x 3 ) = x 1 x 1 + x 2 x 2 x 1 x 2 Solution: nullity( L ) = 1 and rank( L ) = 2 2. Let V be an ndimensional vector space, and let T : V V be defined by...
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This note was uploaded on 09/30/2011 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.
 Spring '08
 CELMIN
 Math

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