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**Unformatted text preview: **4 ) A = - 1 1 0- 1 0 1- 1 1 . Note the sum of the columns is the zero vector so the rank of A is at most 2. It is in fact 2 as the first two columns are visibly dependent. Taking ~ b to be the first column of A , we can solve A~x = ~ b . A vector perpendicular to the first two columns will not be in the column space of A . Playing with numbers a bit (or taking the cross product of the first two columns) gives ~ c = 1- 1 1 as a vector for which there is no solution to A~x = ~ c . For a given ~ b , if we want a solution, ~ b must lie in the plane spanned by the first two columns. this plane is perpendicular to ~ c , so ~ b must be perpendicular to ~ c . 7 ) A T CA = - 1- 1 1- 1 1 1 1 0 0 0 2 0 0 0 2 - 1 1 0- 1 0 1- 1 1 = - 1- 1 1- 1 1 1 - 1 1 0- 2 0 2- 2 2 = 3- 1- 2- 1 3- 2- 2- 2 4 ....

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