Page 309 - Matrix Analysis & Applied Linear Algebra

5.4 Orthogonal Vectors 305

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5.4.13. Consider a real inner-product space, where = .
(a) Prove that if x = y , then (x + y) ⊥ (x − y).
2
(b) For the standard inner product in , draw a picture of this.
That is, sketch the location of x + y and x − y for two vectors
with equal norms.

5.4.14. Pythagorean Theorem. Let V be a general inner-product space in
2
which = .
(a) When V is a real space, prove that x ⊥ y if and only if
2 2 2
x + y = x + y . (Something would be wrong if this
were not true because this is where the definition of orthogonal-
ity originated.)
(b) Construct an example to show that one of the implications in
part (a) does not hold when V is a complex space.
(c) When V is a complex space, prove that x ⊥ y if and only if
2 2 2
αx + βy = αx + βy for all scalars α and β.
5.4.15. Let B = {u 1 , u 2 ,..., u n } be an orthonormal basis for an inner-product

ξ
space V, and let x = i i u i be the Fourier expansion of x ∈V.
(a) If V is a real space, and if θ i is the angle between u i and x,
explain why
ξ i = x cos θ i .
2 3
Sketch a picture of this in or to show why the com-
ponent ξ i u i represents the orthogonal projection of x onto
the line determined by u i , and thus illustrate the fact that a
Fourier expansion is nothing more than simply resolving x into
mutually orthogonal components.
42 n 2 2
(b) Derive Parseval’s identity, which says |ξ i | = x .
i=1
5.4.16. Let B = {u 1 , u 2 ,..., u k } be an orthonormal set in an n-dimensional
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inner-product space V. Derive Bessel’s inequality, which says that
if x ∈V and ξ i = u i x , then
k
2 2
|ξ i | ≤ x .
i=1
Explain why equality holds if and only if x ∈ span {u 1 , u 2 ,..., u k } .
k 2
ξ
Hint: Consider x − i=1 i u i .
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This result appeared in the second of the five mathematical publications by Marc-Antoine
Parseval des Chˆenes (1755–1836). Parseval was a royalist who had to flee from France when
Napoleon ordered his arrest for publishing poetry against the regime.
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This inequality is named in honor of the German astronomer and mathematician Friedrich
Wilhelm Bessel (1784–1846), who devoted his life to understanding the motions of the stars.
In the process he introduced several useful mathematical ideas.