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Unformatted text preview: CS3251, Homework 3 Solution 1. Problem # 10 The following are subspaces: (a), (d), (e) 2. Problem # 23 If we add b to a matrix A then the column space gets larger unless b C ( A ) (i.e., is already a linear combination of the columns of A ). Example, of a larger space ( b / C ( A )) [ A, b ] = 1 0 1 0 0 1 . Example, of the same space ( b C ( A ) and so C ([ A, b ]) = C ( A )) [ A, b ] = 1 0 1 0 1 1 . If the system Ax = b is solvable exactly then b must belong to the column space of A . Otherwise, no linear combination of the columns of A gives b . But b C ( A ) implies that the column space of the matrix [ A, b ] is equal to the column space of A , i.e., C ([ A, b ]) = C ( A ). 3. Problem # 31 If the nullspace of A consists of all the multiples of x = (2 , 1 , , 1) T then we have a single special solution and, therefore, a single free variable. Then the dimension of N ( A ) is n- r = 1, where n denotes the number of columns and in this case n = 4. Thus, we have r = n- 1 = 3 pivots. Assuming that the first three1 = 3 pivots....
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This note was uploaded on 01/31/2012 for the course COMS 3251 taught by Professor Anargyrospapageorgiou during the Fall '11 term at Columbia.
- Fall '11