MATH 330 H43

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

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Unformatted text preview: 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 1 13- 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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This note was uploaded on 09/19/2009 for the course MATH 330 taught by Professor Johnson,j during the Spring '08 term at Nevada.

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

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