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quiz1_1806_sol_s05

# quiz1_1806_sol_s05 - 18.06 Professor Strang Quiz 1 Grading...

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3 18.06 Professor Strang Quiz 1 February 28, 2005 Grading 1 2 Your PRINTED name is: SOLUTIONS 4 1 (26 pts.) Suppose A is reduced by the usual row operations to 1 4 0 2 R = 0 0 1 2 . 0 0 0 0 Find the complete solution (if a solution exists) to this system involving the original A : Ax = sum of the columns of A. Solution The complete solution x = x particular + x nullspace has � ⎡ 1 4 2 � ⎢ � ⎢ 1 1 0 � ⎢ = . x particular � ⎢ x nullspace = x 2 + x 4 1 0 2 � ⎣ 1 0 1 The free variables x 2 and x 4 can take any values. The two special solutions came from the nullspace of R = nullspace of A . The particular solution of 1’s gives Ax = sum of the columns of A . Note : This also gives Rx = sum of columns of R . 1

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2 (18 pts.) Suppose the 4 by 4 matrices A and B have the same column space . They may not have the same columns !
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quiz1_1806_sol_s05 - 18.06 Professor Strang Quiz 1 Grading...

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