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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).
 Fall '05
 HUI
 Math, Linear Algebra, Algebra

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