250 exam one review - 640:250:C2 EXAM 1: REVIEW PROBLEMS...

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Unformatted text preview: 640:250:C2 EXAM 1: REVIEW PROBLEMS SPRING 2010 1. (a) What does it mean to say that the vectors u 1 ,... , u k in R n are linearly independent ? Give the precise definition in one or more full sentences. (b) Are the vectors u 1 = 1 3- 2 , u 2 = 3 5- 3 , and u 3 = 1- 1 1 linearly independent? Justify your answer in terms of the definition you gave in (a). 2. (a) What is meant by the span of a set of vectors S = { u 1 ,..., u k } ? Give the precise definition in one or more full sentences. (b) Suppose that u 1 = 1 3- 2 , u 2 = 3 5- 3 , and u 3 = 1- 1 2 . Is the span of the set of vectors S = { u 1 , u 2 , u 3 } all R 3 ? Justify your answer in terms of the definition you gave in (a). 3. In (a)-(c) below we suppose that we have been given a system of equations A x = b and that we have already reduced the augmented matrix [ A b ] to the reduced row-echelon form [ R c ] given. In each case, determine (i) whether the original equations have a solution; (ii) if they do have a solution, whether or not it is unique; and (iii) if it is not unique, on how many free parameters there...
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250 exam one review - 640:250:C2 EXAM 1: REVIEW PROBLEMS...

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