exam1Solutions

exam1Solutions - Math 222-500 1. Solutions Exam 1 October...

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Math 222-500 Solutions Exam 1 October 10, 2006 1 . (15) Define the following: a . the span of the set of vectors x u 1 , x u 2 , u , x u k , The set of all linear combinations of these vectors is the span. b . the set of vectors x u 1 , x u 2 , u , x u k is linearly independent. The set is linearly independent if whenever c 1 x u 1 U u U c k x u k ± 0 u then each of the c i must equal zero. c . the null space of an m u n matrix A . The null space of A is the set of vectors x u u R n such that Ax u ± 0 u . 2 . (25) A system of equations has A as its coefficient matrix, and A is row equivalent to the matrix B ± 1 0 2 1 0 1 0 1 U 1 0 0 2 0 0 0 0 1 1 0 0 0 0 0 0 . a . Find a basis for the null space of A . The matrix B tells us that x 5 ± U x 6 , x 2 ± x 3 U 2 x 6 , and x 1 ± U 2 x 3 U x 4 U x 6 . That is, we have 3 bound variables ( x 1 , x 2 , and x 5 ) with the other 3 being free variables. Thus, the null space has dimension 3, and basis equal to U U 2,1,1,0,0,0 ± , U U 1,0,0,1,0,0 ± , U U 1, U 2,0,0, U 1,1 ± . b . What is the dimension of the column space of A ? The dimension of the column space is the number of free variables, which is 3. c . What is the dimension of the row space of A ? The dimension of the row space is the same as the dimension of the column space. Thus, it too is 3. d
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exam1Solutions - Math 222-500 1. Solutions Exam 1 October...

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