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sample_tt2_1 - Math 136 Sample Term Test 2 1 NOTES In...

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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. Also, our test covers change of coordinates. 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 = a 1 1 d M (2 , 2) a + d = 0 of M (2 , 2). c) U = p ( x ) P 2 p (2) = 0 of P 2 .
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