Page 444 - Matrix Analysis & Applied Linear Algebra

440 Chapter 5 Norms, Inner Products, and Orthogonality

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5.13.5. Explain why P M = whenever B = {u 1 , u 2 ,..., u r } is an
i=1 u i u i
n×1
orthonormal basis for M⊆ .
5.13.6. Explain how to use orthogonal reduction techniques to compute the
orthogonal projectors onto each of the four fundamental subspaces of a
m×n
matrix A ∈ .

5.13.7. (a) Describe all 2 × 2 orthogonal projectors in 2×2 .
(b) Describe all 2 × 2 projectors in 2×2 .

n
5.13.8. The line L in passing through two distinct points u and v is
L = u + span {u − v} . If u = 0 and v = αu, then L is a line not
passing through the origin—i.e., L is not a subspace. Sketch a picture
2
3
in or to visualize this, and then explain how to project a vector
b orthogonally onto L.

5.13.9. Explain why : x is a least squares solution for Ax = b if and only if

A: x − b = P N(A T ) b
.

2 2
5.13.10. Prove that if ε = A: x − b, where : x is a least squares solution for
2 2

2
Ax = b, then ε = b − P R(A) b
.

2 2 2
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5.13.11. Let M be an r-dimensional subspace of . We know from (5.4.3)
that if B = {u 1 , u 2 ,..., u r } is an orthonormal basis for M, and if
x ∈M, then x is equal to its Fourier expansion with respect to B.
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That is, x = (u i x)u i . However, if x /∈M, then equality is not
i=1
possible (why?), so the question that arises is, “What does the Fourier
expansion on the right-hand side of this expression represent?” Answer
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this question by showing that the Fourier expansion (u i x)u i is
i=1
the point in M that is closest to x in the euclidean norm. In other
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words, show that (u i x)u i = P M x.
i=1
5.13.12. Determine the orthogonal projection of b onto M, where
       
5 −3/5 0 4/5
 
 
 2   0   0   0 
5  4/5 0 3/5 
b =   and M = span   ,   ,   .
 
3 0 1 0
Hint: Is this spanning set in fact an orthonormal basis?