Prelim 2 [Solutions]

B true we show they are independent then their span

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Unformatted text preview: ALSE. The null space of A is a subspace of R3 while the the null space of AT is a subspace of R2 . It is meaningless to speak of them being perpendicular. b) TRUE. We show they are independent. Then their span is a three dimensional subspace of R3 which is necessarily all of R3 . Suppose c1￿1 + c2￿2 +3 ￿3 = ￿ Dotting with ￿1 we get v v v 0. v c1 = 0. Similarly, dotting with ￿2 and ￿3 gives c2 = 0 and c3 = 0. v v c) TRUE. Take any basis and apply Gram-Schmidt. d) TRUE. The 3 × 3 matrix ￿ ￿ T has rank at most the rank of the 3 × 1 matrix ￿ which is uv u T at most 1. Thus ￿ ￿ is not inve...
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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.

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