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MATH 330 H46

# MATH 330 H46 - Math 330 Homework 4.6(Pages 269,270(4 The...

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Math 330 Homework 4.6 (Pages 269,270) (4) The matrix B is row equivalent to A . Without calculations, list rank A and dim Nul A . Then find bases for Col A , Row A, and Nul A . A = 1 1 - 3 7 9 - 9 1 2 - 4 10 13 - 12 1 - 1 - 1 1 1 - 3 1 - 3 1 - 5 - 7 3 1 - 2 0 0 - 5 - 4 B = 1 1 - 3 7 9 - 9 0 1 - 1 3 4 - 3 0 0 0 1 - 1 - 2 0 0 0 0 0 0 0 0 0 0 0 0 SOLUTION: B has three pivots in positions (1 , 1) , (2 , 2), and (3 , 4). This means that dim Col A or the rank of A is 3. These matrices have 6 columns so dim Nul A is 6 - 3 = 3. B is in Echelon form so its nonzero rows form a basis for Row A : 1 { [1 1 - 3 7 9 - 9] , [0 1 - 1 3 4 - 3] , [0 0 0 1 - 1 - 2] } The pivot columns (which are columns 1,2, and 4) tell us that these columns in A form a basis for Col A : 1 1 1 1 1 , 1 2 - 1 - 3 - 2 , 7 10 1 - 5 0 To find a dimension for Nul A we continue row reductions of B to reduced Echelon form: ( R 1 ,R 2) ( R 1 - 7 R 3 ,R 2 - 3 R 3) = 1 1 - 3 0 16 5 0 1 - 1 0 7 3 0 0 0 1 - 1 - 2 0 0 0 0 0 0 0 0 0 0 0 0 R 1 R 1 - R 2 = 1 0 - 2 0 9 2 0 1 - 1 0 7 3 0 0 0 1 - 1 - 2 0 0 0 0 0 0 0 0 0 0 0 0 x 1 = 2 x 3 - 9 x 5 - 2 x 6 x 2 = x 3 - 7 x 5 - 3 x 6 x 4 = x 5 + 2 x 6 x 1 x 2 x 3 x 4 x 5 x 6 =

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MATH 330 H46 - Math 330 Homework 4.6(Pages 269,270(4 The...

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