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Unformatted text preview: Math 235 1. Short Answer Problems Sample Term Test 1  1 a) Give the deﬁnition of an inner product , on a vector space V . b) Let B = {v1 , . . . , vn } be orthonormal in an inner product space V and let v ∈ V such that v = a1 v1 + · · · + an vn . Prove that ai =< v, vi >. c) Deﬁne what it means for a set B to be orthonormal in an inner product space V . d) State the RankNullity Theorem. e) Find the rank and nullity of the linear mapping L : R3 → M (2, 2) deﬁned by L(x1 , x2 , x3 ) = x1 x1 + x2 . x2 x1 − x2 2. Let V be an ndimensional vector space, and let T : V → V be deﬁned by T (v ) = λv for all v in V , where λ ∈ R is a constant. a) Prove that T is linear. b) Compute the nullspace and the range of T . There are two cases, depending on λ. c) Let β = {v1 , . . . , vn } be a basis for V . Give the matrix [T ]β for the map T with respect to the basis β . 3. Find the matrix of L : R2 → P2 deﬁned by L(a1 , a2 ) = a1 x2 + (a1 + a2 ) with respect 1 1 to the basis B = , of R2 and C = {x2 + 1, x + 1, x2 − x − 1} of P2 . −1 2 4. Let A be an m × n matrix. Prove that Ax = b is consistent for all b ∈ Rm if and only if the equation AT y = 0 has only the trivial solution. 5. Let V be a vector space of dimension n. Prove that the vector space of all linear operators from V to V is isomorphic to M (n, n). 6. Let < , > be the standard inner product in Rn and let U be an m × n matrix with orthonormal columns. Let x, y ∈ Rn . Prove that < x, y >=< U x, U y > and thus that Ux = x . 7. Let B = 2 1 , 3 2 and C = 2 1 , 1 1 and deﬁne L : R2 → R2 such that [x]B = [L(x)]C . a) Find L b) Find L 3 . 5 x1 . x2 c) Prove that L is an isomorphism. 8. Let B = 1 −2 , 0 1 < ∈ R2 and deﬁne x x1 , 2 >= x1 x2 + 8y1 y2 + 2x1 y2 + 2y1 x2 . y2 y1 a) Prove that < , > deﬁnes an inner product on R2 . b) Show that B is an orthogonal basis for R2 using this inner product and produce an orthonormal basis. c) Find the B coordinates of 2 3 ...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Vector Space

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