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S ⊆ R , prove that (I − P ) = I − P , where I is the n × n identity matrix.
(b) (8 points) Give an example of a 3 × 3 matrix A whose columns are orthonormal and A
has exactly one entry equal to 0.
Solutions: (a) Since P is a projection matrix we know P 2 = P . So (I − P )2 = (I − P )(I −
P ) = I 2 − P I − IP − P 2 = I − 2P + P 2 = I − 2P + P = I − P .
(b) We’ll deal with the length 1 stuﬀ at the end. First we get the ‘ortho’ stuﬀ down. Let the
ﬁrst vector be in the xy plane. It’ll have third component equal to zero. So (1, 1, 0) will do.
The next vector has to be (a, b, c) where none of the entires are 0 and...
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 Fall '05
 HUI
 Linear Algebra, Algebra, Vectors

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