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# assign7 - Math 136 Assignment 7 Due Wednesday Mar 10th 1...

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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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