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MATH 330 H43 - Math 330 Homework 4.3(Pages 243,244,245 For...

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Math 330 Homework 4.3 (Pages 243,244,245) For problems (4) and (6) decide if the collection is (i) linearly independent and/or (ii) spans R 3 and/or (iii) is a basis for R 3 (4) 2 - 2 1 , 1 - 3 2 , - 7 5 4 SOLUTION: Since there are 3 vectors, they will span R 3 iff they are linearly independent iff they form a basis for R 3 . To see if they do, we make them the columns of a single matrix which we can row reduce: A = 2 1 - 7 - 2 - 3 5 1 2 4 R 1 R 3 = 1 2 4 - 2 - 3 5 2 1 - 7 ( R 2 ,R 3) ( R 2+2 R 1 ,R 3 - 2 R 1) = 1 2 4 0 1 13 0 - 3 - 15 R 3 R 3+3 R 2 = 1 2 4 0 1 13 0 0 24 This is Echelon form. We have three pivots and so the vectors are linearly independent, they span | bfR 3 , and they form a basis for R 3 . (6) 1 2 - 3 , - 4 - 5 6 SOLUTION: Since there are only two vectors we know that they cannot span R 3 (they can only span a 2-dimensional plane within R 3 ) and therefore they cannot be a basis for R 3 . Furthermore, it is always easy to see if two vectors are linearly independent - we just need to see if they are multiples of each other. They are not multiples of each other and hence they are linearly independent.
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