Page 446 - Matrix Analysis & Applied Linear Algebra

442 Chapter 5 Norms, Inner Products, and Orthogonality

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5.13.17. An affine space v + M⊆ for which dim M = n − 1is called a
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hyperplane. For example, a hyperplane in is a line (not necessarily
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through the origin), and a hyperplane in is a plane (not necessarily
through the origin). The i th equation A i∗ x = b i in a linear system
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A m×n x = b is a hyperplane in , so the solutions of Ax = b occur
at the intersection of the m hyperplanes defined by the rows of A.
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(a) Prove that for a given scalar β and a nonzero vector u ∈ ,
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the set H = {x | u x = β} is a hyperplane in .
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(b) Explain why the orthogonal projection of b ∈ onto H is

T
T
p = b − u b − β/u u u.
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5.13.18. For u, w ∈ such that u w =0, let M = u ⊥ and W = span {w} .
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(a) Explain why = M⊕W.
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(b) For b ∈ , explain why the oblique projection of b onto
T
T
M along W is given by p = b − u b/u ww.
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(c) For a given scalar β, let H be the hyperplane in defined
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by H = {x | u x = β}—see Exercise 5.13.17. Explain why the
oblique projection of b onto H along W should be given by

T
T
p = b − u b − β/u w w.
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5.13.19. Kaczmarz’s Projection Method. The solution of a nonsingular
system

a 11 a 12 x 1 b 1
=
a 21 a 22 x 2 b 2
is the intersection of the two hyperplanes (lines in this case) defined by
H 1 ={(x 1 ,x 2 ) | a 11 x 1 +a 12 x 2 = b 1 } , H 2 ={(x 1 ,x 2 ) | a 21 x 1 +a 22 x 2 = b 2 }.

It’s visually evident that by starting with an arbitrary point p 0 and
alternately projecting orthogonally onto H 1 and H 2 as depicted in
Figure 5.13.7, the resulting sequence of projections {p 1 , p 2 , p 3 , p 4 ,... }
converges to H 1 ∩H 2 , the solution of Ax = b.

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Although this idea has probably occurred to many people down through the ages, credit is
usually given to Stefan Kaczmarz, who published his results in 1937. Kaczmarz was among a
school of bright young Polish mathematicians who were beginning to flower in the first part
of the twentieth century. Tragically, this group was decimated by Hitler’s invasion of Poland,
and Kaczmarz himself was killed in military action while trying to defend his country.