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sample_tt2_2 - Math 136 Sample Term Test 2 2 NOTES In...

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Math 136 Sample Term Test 2 # 2 NOTES: - In addition to these questions you should also do questions 7, 10 b from sample term test 1 # 2. 1. Short Answer Problems a) Let S = { v 1 , . . . , v n } be a non-empty subset of a vector space V . Define the statement S is linearly independent. b) Write the definition of a subspace S of a vector space V . c) Write the definition of the dimension of a vector space V . d) Prove that 0 x = 0 for any x V . e) Is it true that if a set S with more than one vector is linearly dependent then every vector v S can be written as a linear combination of the other vectors. Justify your answer. 2. Let β = { x 2 - 4 x + 4 , x - 2 , 1 } . a) Show that span( β ) = P 2 . b) Let w = x 2 + x + 1. Find the β coordinate vector of w . 3. Determine, with proof, which of the following are subspaces of the given vector space. a) S = { ax 2 + bx + c | b 2 - 4 ac = 0 } of P 2 . b) D = a b c d ad - bc = 0 of M (2 , 2). c) A = a b c d a + b + c + d = 0 of M (2 , 2). 4. Let A = 3 6 1 1 2 4 2 4 3 and let L be a linear mapping with matrix A . a) Find a basis for the nullspace of L . b) Find a basis for the range of L . 5. Find a basis for the following subspaces of R 3 and state the dimension of the subspace. a) span { (2 , 4 , 6) , (4 , 5 , 6) , (1 , 1 , 1) } b) The plane 2
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