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Unformatted text preview: Math 136 Assignment #3
1. Let A be a 3 4 matrix, and b be a vector in R3 . Let 1 2 A = b, 0 2 and A be row equivalent to the matrix 0 1 5 8 0 0 1 2 . 0 3 0 1 Find all solutions to Ax = b. 1 1 3 2 2. Let v1 = 2, v2 = 1 , v3 = 3 and v4 = 4. 3 2 8 1 (a) Determine whether or not the set S = {v1 , v2 , v3 , v4 } is linearly independent. (b) Find a subset of S consisting of three linearly independent vectors. 1 0 1 3. Find all values of a for which the vectors 2, 1 and 2a  2 are linearly dependent. 3 1 1 4. Let f1 (x) = x2 + 2x + 3, f2 (x) = x2 + x + 2, f3 (x) = x2 + 3x + 4 and f4 (x) = x2 + 1. Write one of these polynomials in terms of the others. a b , and ad  bc = 0. Prove that the columns of A are linearly independent. c d 5. Let A = 6. Let {v1 , v2 , v3 , v4 } be a linearly independent set of vectors. Determine whether or not the following sets are linearly independent: (a) {v1 , v1 + v2 , v1 + v2 + v3 , v1 + v2 + v3 + v4 } (b) {v1  v2 , v2  v3 , v3  v4 , v4  v1 } 7. Let S and T be sets of linearly independent vectors in Rn . (a) The union of S and T , denoted S T , is defined to be the set of vectors in S or T . Show, by providing an example, that S T can be linearly dependent. (b) The intersection of span(S) and span(T ), denoted span(S) span(T ), is defined to be the set of vectors in span(S) and span(T ). Prove that if span(S) span(T ) contains only the zero vector, then S T is linearly independent. ...
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This note was uploaded on 07/01/2008 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.
 Spring '08
 All
 Math, Linear Algebra, Algebra

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