Page 310 - Matrix Analysis & Applied Linear Algebra

306 Chapter 5 Norms, Inner Products, and Orthogonality

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5.4.17. Construct an example using the standard inner product in to show
that two vectors x and y can have an angle between them that is close
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to π/2 without x y being close to 0. Hint: Consider n to be large,
and use the vector e of all 1’s for one of the vectors.

5.4.18. It was demonstrated in Example 5.4.3 that y is linearly correlated with
x in the sense that y ≈ β 0 e + β 1 x if and only if the standardization
vectors z x and z y are “close” in the sense that they are almost on the
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same line in . Explain why simply measuring z x − z y does not
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always gauge the degree of linear correlation.
5.4.19. Let θ be the angle between two vectors x and y from a real inner-
product space.
(a) Prove that cos θ =1 if and only if y = αx for α> 0.
(b) Prove that cos θ = −1if and only if y = αx for α< 0.
Hint: Use the generalization of Exercise 5.1.9.

5.4.20. With respect to the orthonormal set
!
1 cos t cos 2t sin t sin 2t sin 3t
B = √ , √ , √ ,..., √ , √ , √ ,... ,
2π π π π π π
determine the Fourier series expansion of the saw-toothed function
defined by f(t)= t for −π< t < π. The periodic extension of this
function is depicted in Figure 5.4.3.

π

−π π

−π

Figure 5.4.3