sample_tt1_1

# sample_tt1_1 - Math 235 Sample Midterm 1 NOTES Our Term...

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Unformatted text preview: Math 235 Sample Midterm - 1 NOTES: - Our Term Test 1 does not cover the Gram-Schmidt procedure. 1. Short Answer Problems a) Give the definition of an inner product h , i on a vector space V . 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 = h ~v,~v i i . c) Define what it means for a set B to be orthonormal in an inner product space V . d) State the Rank-Nullity Theorem. 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 . 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. a) Prove that T is linear. b) Compute the kernel and the range of T . There are two cases, depending on λ . c) Let B = { ~v 1 ,...,~v n } be a basis for V . Give the matrix [ T ] B for the map T with respect to the basis B ....
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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.

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sample_tt1_1 - Math 235 Sample Midterm 1 NOTES Our Term...

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