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quiz2-sol

# quiz2-sol - augmented matrix a 1 a 2 a 3 | b is consistent...

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Name: MATH227 Quiz 2 (Fall 2006) For full credit, please show all steps in details!! Let A = [ a 1 a 2 a 3 ] = 2 0 6 - 1 8 5 1 - 2 1 , b = 10 3 3 , and W be the set of all linear combinations of the columns of A (i.e., W = Span { a 1 , a 2 , a 3 } ). (a) Show that 0 and a 2 are in W . (4 points) (b) Is the vector b in W ? How about the vector a 1 + b ? (6 points) (a) Since 0 = 0 · a 1 + 0 · a 2 + 0 · a 3 and a 2 = 0 · a 1 + 1 · a 2 + 0 · a 3 , the vectors 0 and a 2 are in W = Span { a 1 , a 2 , a 3 } . (b) Consider the augmented matrix [ a 1 a 2 a 3 | b ] = 2 0 6 | 10 - 1 8 5 | 3 1 - 2 1 | 3 R 2 + 1 2 R 1 R 3 - 1 2 R 1 2 0 6 | 10 0 8 8 | 8 0 - 2 - 2 | - 2 R 3 + 1 4 R 2 2 0 6 | 10 0 8 8 | 8 0 0 0 | 0 . We see from the above row echelon form that the system of linear equations with
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Unformatted text preview: augmented matrix [ a 1 a 2 a 3 | b ] is consistent. Then, it is equivalent to say that b is a linear combination of a 1 , a 2 , a 3 . That is, b is in W . Now, as b is in W , b = c 1 a 1 + c 2 a 2 + c 3 a 3 for some scalars c 1 , c 2 , c 3 . Then a 1 + b = ( c 1 + 1) a 1 + c 2 a 2 + c 3 a 3 . Therefore, a 1 + b is also in W . 1...
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