123 29 the matrix 2 0 0 has 1 2 3 in both the row

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Unformatted text preview: −1 −5 0 0 42 The product is diagonal because it’s oﬀ diagonal entries are dot products of diﬀerent columns of A, which are 0 by construction. 123 29) The matrix 2 0 0 has (1, 2, 3) in both the row and column space. 300 1 0 −1/3 The matrix 2 0 −2/3 has (1, 2, 3) in both the nullspace and column space. 3 0 −1 Can a (nonzero) vector ￿ be in both the column space of A and the nullspace of AT ? No, b these spaces are orthogonal and no nonzero vector is perpendicular to itself. We have dealt with all pairs involving the column space. Similarly the nullspace of A is orthogonal to the row space of A so they share no nonzero vector. Can (1, 2, 3) be in the row space of A and the nullspace of AT ? Sure take T 1 0 −1/3 1 23 0 0 0 . A = 2 0 −2/3 = 3 0 −1 −1/30 −2/3 −1 Finally, can (1, 2, 3) be in the nullspace of A and the nullspace of AT ? Sure take A to be the zero matrix!...
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This note was uploaded on 01/30/2014 for the course MATH 2940 taught by Professor Hui during the Fall '05 term at Cornell University (Engineering School).

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