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sample_tt2_1

# sample_tt2_1 - a span(1-2 1(2 10(0 1 2(1 1 7 b The...

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Math 136 Sample Term Test 2 # 1 NOTES: - In addition to these questions you should also do questions 4d, 5, 6 from sample term test 1 # 1. 1. Short Answer Problems a) What is the deﬁnition of the row space and column space of a matrix A . b) What is the deﬁnition of a subspace. c) What is the deﬁnition of a basis. d) Let B = { (1 , 2 , 1) , ( - 1 , 0 , 2) , (1 , 1 , 1) } . If [ ~v ] B = 1 - 1 1 what is ~v ? e) Let V be a vector space and ~v V . Prove that ( - 1) ~v is the additive inverse of ~v . 2. Let β = { 1 + x 2 , 1 + x + x 2 , 1 + 2 x + 2 x 2 } . a) Show that β is a basis for P 2 . b) Find the β coordinates of the standard basis vectors of P 2 . 3. Determine, with proof, which of the following are subspaces of the given vector space. a) S = { ( x 1 ,x 2 ,x 3 ,x 4 ) R 4 ± ± ± ± 2 x 1 - 5 x 4 = 0 and 3 x 2 - 2 x 4 = 0 } of R 4 . b) T = ²³ a 1 1 d ´ M (2 , 2) ± ± ± ± a + d = 0 µ of M (2 , 2). c) U = ² p ( x ) P 2 ± ± ± ± p (2) = 0 µ of P 2 . 4. Let R : R 3 R 3 be a reﬂection in the plane x + 2 y + z = 0. Find a basis for the null space and range of R . 5. Find a basis for the following subspaces and state the dimension of the subspace.
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Unformatted text preview: a) span { (1 ,-2 , 1) , (2 , , 10) , (0 , 1 , 2) , (1 , 1 , 7) } b) The hyperplane x 1 + 2 x 2-x 3 + x 4 = 0. 6. Let V and W be vector spaces and let T : V → W be a linear mapping. a) Let { ~v 1 ,...,~v k } be a non-empty linearly independent set in V . Prove that if the null space of T is { ~ } then { T ( ~v 1 ) ,...,T ( ~v k ) } is also linearly independent. b) Suppose { ~v 1 ,...,~v k } is a spanning set for V and that the range of T is W . Prove that { T ( ~v 1 ) ,...,T ( ~v k ) } is a spanning set for W . c) Assume that dim V = n and dim W = n . Prove that the null space of T is { ~ } if and only if the range of T is W . 7. State and prove the subspace test....
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