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sample_tt1_1_ans - Math 235 Sample Midterm 1 Answers NOTE...

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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 ~ 0 , ~ 0 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 Rank-Nullity Theorem. Solution: Suppose that V is an n -dimensional 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 n -dimensional vector space, and let T : V V be defined by T ( ~v ) = λ~v for all ~v in V , where λ R is a constant.
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