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assign7 - ² 2 1 0 1 ³ 6 Suppose that B = ~v 1,~v k is an...

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Math 136 Assignment 7 Due: Wednesday, Mar 10th 1. Show the each of the following sets form a basis for the subspace that they span, and determine the coordinates of x and y with respect to the basis. a) { (1 , 1 , 0 , 1 , 0) , (1 , 0 , 2 , 1 , 1) , (0 , 0 , 1 , 1 , 3) } ; x = (2 , - 2 , 5 , - 1 , - 5), y = ( - 1 , - 3 , 3 , - 2 , - 1). b) 1 1 1 0 , 0 1 1 1 , 2 0 0 - 1 ; x = 0 1 1 2 , y = - 4 1 1 4 . 2. Find a basis and determine the dimension of the following sets. a) S = span { (1 , 2 , 1) , (2 , 3 , - 2) , ( - 1 , 0 , 7) , (2 , 2 , 1) } . b) S = span { 1 + x + x 2 , x + x 2 + x 3 , 1 + x 2 + x 3 } . 3. Find a basis for the hyperplane x 1 - x 2 + x 3 + 2 x 4 = 0 in R 4 and then extend the basis to obtain a basis for R 4 . 4. Find a basis for R 3 that includes the vectors (1 , 2 , - 1) and (3 , - 1 , 1). 5. Find a basis for M (2 , 2) that includes the vectors v 1 = 1 2 1 1 , v 2 = 2 1 0 1 . 6. Suppose that
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Unformatted text preview: ² 2 1 0 1 ³ . 6. Suppose that B = { ~v 1 ,...,~v k } is an ordered basis for a vector space V , and that C = { ~w 1 ,..., ~w k } is another ordered basis for V and that [ ~x ] B = [ ~x ] C for all ~x ∈ V . Prove that ~v i = ~w i for 1 ≤ i ≤ k . 7. Let V be a ﬁnite dimensional vector space and let U be a subspace of V . a) Prove that dim U ≤ dim V . b) Prove that if W is a subspace of V such that W ⊆ U , then dim W ≤ dim U . c) Prove that if W ⊆ U and dim W = dim U , then W = U . 1...
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