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fall1997-final

# fall1997-final - Name MAS 3114 Computational Linear Algebra...

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Name: MAS 3114 Computational Linear Algebra Fall 1997 FINAL EXAM Instructions: There are 12 questions. There 223 points available. Full marks (100%) will be given for 210 points. Write on ONE side of the paper provided. Show all necessary working and reasoning to receive full credit. Your work needs to be written in a proper and coherent fashion. When giving proofs your reasoning should be clear. Calculators are allowed (for simple calculations). Questions with may be answered on the exam paper. 1

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1. [4 + 10 + 4 = 18 points] (a) (i) Let v 1 , v 2 , . . . , v p be vectors in R n . Define the set denoted by Span { v 1 , v 2 , . . . , v p } . (b) Let v 1 = 1 4 - 2 , v 2 = 5 19 4 , v 3 = 3 8 - 19 . Let W = Span { v 1 , v 2 } . (i) Determine if v 3 is in the subspace W , showing working. Use PART 1 of the MATLAB HANDOUT to check your answer. (ii) Find dim W giving reason. 2. [4 points] Complete the following definition : A set { v 1 , v 2 , . . . , v p } in R n is said to be linearly independent if the vector equation has . 3. [5 × 4 = 20 points] Let v 1 = 1 - 3 - 3 , v 2 = - 1 4 1 , v 3 = 0 1 - 2 , v 4 = 0 0 0 , v 5 = 1 2 3 , v 6 = v 1 - 2 v 3 , v 7 = 3 2 1 . Determine which of the following sets of vectors are linearly independent giving reasons. You might find PART 2 of the MATLAB HANDOUT useful. (i) { v 1 , v 2 , v 3 } , (ii) { v 2 , v 3 , v 4 } , (iii) { v 3 , v 5 } , (iv) { v 1 , v 3 , v 5 , v 7 } , (v) { v 1 , v 3 , v 6 } .
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