Math Problems Vault

Problem 5.4.1

Using the standard inner product, determine which of the following pairs are orthogonal vectors in the indicated space.

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(a)

$x = \begin{pmatrix} 1 \\ -3 \\ 4 \end{pmatrix}$ and $y = \begin{pmatrix} -2 \\ 2 \\ 2 \end{pmatrix}$ in $\mathbb {R}^{3}$ ,

(b)

$x = \begin{pmatrix} i \\ 1+i \\ 2 \\ 1-i \end{pmatrix}$ and $y = \begin{pmatrix} 0 \\ 1+i \\ -2 \\ 1-i \end{pmatrix}$ in $\mathbb {C}^{4}$ ,

(c)

$x = \begin{pmatrix} 1 \\ -2 \\ 3 \\ 4 \end{pmatrix}$ and $y = \begin{pmatrix} 4 \\ 2 \\ -1 \\ 1 \end{pmatrix}$ in $\mathbb {R}^{4}$ ,

(d)

$x = \begin{pmatrix} 1+i \\ 1 \\ i \end{pmatrix}$ and $y = \begin{pmatrix} 1-i \\ -3 \\ -i \end{pmatrix}$ in $\mathbb {C}^{3}$ ,

(e)

$x = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix}$ and $y = \begin{pmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{n} \end{pmatrix}$ in $\mathbb {R}^{n}$ .

Problem 5.4.2

Find two vectors of unit norm that are orthogonal to $u = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ .

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Problem 5.4.3

(a)(b)(c)

Consider the following set of three vectors:

x_{1} = \begin{pmatrix} 1 \\ -1 \\ 0 \\ 2 \end{pmatrix}, \quad x_{2} = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 0 \end{pmatrix}, \quad x_{3} = \begin{pmatrix} -1 \\ -1 \\ 2 \\ 0 \end{pmatrix}.

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(a)

Using the standard inner product in $\mathbb {R}^{4}$ , verify that these vectors are mutually orthogonal.

(b)

Find a nonzero vector $x_{4}$ such that $\left\{ x_{1},x_{2},x_{3},x_{4}\right\}$ is a set of mutually orthogonal vectors.

(c)

Convert the resulting set into an orthonormal basis for $\mathbb {R}^{4}$ .

Problem 5.4.4

Using the standard inner product, determine the Fourier expansion of $x$ with respect to $B$ , where

x = \begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}, \qquad B = \left\{ \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, \; \frac{1}{\sqrt{3}}\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \; \frac{1}{\sqrt{6}}\begin{pmatrix} -1 \\ -1 \\ 2 \end{pmatrix}\right\} .

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Problem 5.4.5

With respect to the inner product for matrices $\left\langle A, B \right\rangle = \operatorname {trace}(A^{T}B)$ , verify that the set

B = \left\{ \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \; \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \; \frac{1}{2}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}, \; \frac{1}{2}\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}\right\}

is an orthonormal basis for $\mathbb {R}^{2 \times 2}$ , and then compute the Fourier expansion of $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ with respect to $B$ .

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Problem 5.4.6

Determine the angle between $x = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $y = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ .

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Problem 5.4.7

Given an orthonormal basis $B$ for a space $V$ , explain why the Fourier expansion for $x \in V$ is uniquely determined by $B$ .

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Problem 5.4.8

Explain why the columns of $U_{n \times n}$ are an orthonormal basis for $\mathbb {C}^{n}$ if and only if $U^{*} = U^{-1}$ .

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Remark.

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Such matrices are said to be unitary; their properties are studied in a later section.

Problem 5.4.9

Matrices with the property $A^{*}A = AA^{*}$ are said to be normal (hermitian matrices as well as real symmetric matrices are included in this class). Prove that if $A$ is normal, then $\mathcal{R}(A) \perp \mathcal{N}(A)$ — i.e., every vector in the range of $A$ is orthogonal to every vector in the nullspace of $A$ .

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Remark.

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For $A \in \mathbb {C}^{m \times n}$ , equations (4.5.5)-(4.5.6) state $\mathcal{R}(A^{*}A) = \mathcal{R}(A^{*})$ , $\mathcal{R}(AA^{*}) = \mathcal{R}(A)$ , $\mathcal{N}(A^{*}A) = \mathcal{N}(A)$ , and $\mathcal{N}(AA^{*}) = \mathcal{N}(A^{*})$ .

Problem 5.4.10

(a)(b)

Using the trace inner product $\left\langle A, B \right\rangle = \operatorname {trace}(A^{T}B)$ , determine the angle between the following pairs of matrices.

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(a)

$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ .

(b)

$A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & -2 \\ 2 & 0 \end{pmatrix}$ .

Problem 5.4.11

Why is the definition for $\cos \theta = \left\langle x, y \right\rangle /\left\| x\right\| \left\| y\right\|$ not good for $\mathbb {C}^{n}$ ? Explain how to define $\cos \theta$ so that it makes sense in $\mathbb {C}^{n}$ .

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Problem 5.4.12

If $\left\{ u_{1}, u_{2}, \ldots , u_{n}\right\}$ is an orthonormal basis for an inner-product space $V$ , explain why

\left\langle x, y \right\rangle = \sum _{i} \left\langle x, u_{i} \right\rangle \left\langle u_{i}, y \right\rangle

holds for every $x, y \in V$ .

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Problem 5.4.13

(a)(b)

Consider a real inner-product space, where $\left\| \star \right\|^{2} = \left\langle \star , \star \right\rangle$ .

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(a)

Prove that if $\left\| x\right\| = \left\| y\right\|$ , then $(x+y) \perp (x-y)$ .

(b)

For the standard inner product in $\mathbb {R}^{2}$ , draw a picture of this. That is, sketch the location of $x+y$ and $x-y$ for two vectors with equal norms.

Problem 5.4.14

(a)(b)(c)

Let $V$ be a general inner-product space in which $\left\| \star \right\|^{2} = \left\langle \star , \star \right\rangle$ .

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(a)

When $V$ is a real space, prove that $x \perp y$ if and only if $\left\| x+y\right\|^{2} = \left\| x\right\|^{2} + \left\| y\right\|^{2}$ .

(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 $x \perp y$ if and only if $\left\| \alpha x + \beta y\right\|^{2} = \left\| \alpha x\right\|^{2} + \left\| \beta y\right\|^{2}$ for all scalars $\alpha$ and $\beta$ .

Problem 5.4.15

(a)(b)

Let $B = \left\{ u_{1}, u_{2}, \ldots , u_{n}\right\}$ be an orthonormal basis for an inner-product space $V$ , and let $x = \sum_{i} \xi_{i}u_{i}$ be the Fourier expansion of $x \in V$ .

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(a)

If $V$ is a real space, and if $\theta_{i}$ is the angle between $u_{i}$ and $x$ , explain why $\xi_{i} = \left\| x\right\| \cos \theta_{i}$ . Sketch a picture of this in $\mathbb {R}^{2}$ or $\mathbb {R}^{3}$ to show why the component $\xi_{i}u_{i}$ 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.

(b)

Derive Parseval's identity, which says $\sum_{i=1}^{n} \left|\xi_{i}\right|^{2} = \left\| x\right\|^{2}$ .

Problem 5.4.16

Let $B = \left\{ u_{1}, u_{2}, \ldots , u_{k}\right\}$ be an orthonormal set in an $n$ -dimensional inner-product space $V$ . Derive Bessel's inequality, which says that if $x \in V$ and $\xi_{i} = \left\langle u_{i}, x \right\rangle$ , then

\sum _{i=1}^{k} \left|\xi _{i}\right|^{2} \leq \left\| x\right\| ^{2}.

Explain why equality holds if and only if $x \in \left\langle u_{1}, u_{2}, \ldots , u_{k}\right\rangle$ .

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Problem 5.4.17

Construct an example using the standard inner product in $\mathbb {R}^{n}$ to show that two vectors $x$ and $y$ can have an angle between them that is close to $\pi /2$ without $x^{T}y$ being close to $0$ .

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Problem 5.4.18

Example 5.4.3 shows that $y$ is linearly correlated with $x$ in the sense that $y \approx \beta_{0}e + \beta_{1}x$ if and only if the standardization vectors $z_{x} = (x-\mu_{x}e)/\sigma_{x}$ and $z_{y} = (y-\mu_{y}e)/\sigma_{y}$ are "close" in the sense that they are almost on the same line in $\mathbb {R}^{n}$ (where $\mu_{x}, \sigma_{x}$ are the mean and standard deviation of $x$ 's entries, and similarly for $y$ ). Explain why simply measuring $\left\| z_{x}-z_{y}\right\|_{2}$ does not always gauge the degree of linear correlation.

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Problem 5.4.19

(a)(b)

Let $\theta$ be the angle between two vectors $x$ and $y$ from a real inner-product space.

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(a)

Prove that $\cos \theta = 1$ if and only if $y = \alpha x$ for $\alpha > 0$ .

(b)

Prove that $\cos \theta = -1$ if and only if $y = \alpha x$ for $\alpha < 0$ .

Problem 5.4.20

With respect to the orthonormal set

B = \left\{ \frac{1}{\sqrt{2\pi }}, \; \frac{\cos t}{\sqrt{\pi }}, \; \frac{\cos 2t}{\sqrt{\pi }}, \ldots , \; \frac{\sin t}{\sqrt{\pi }}, \; \frac{\sin 2t}{\sqrt{\pi }}, \; \frac{\sin 3t}{\sqrt{\pi }}, \ldots \right\} ,

determine the Fourier series expansion of the saw-toothed function defined by $f(t) = t$ for $-\pi < t < \pi$ (extended periodically).

$The periodic extension of the saw-toothed function f(t)=t on (-\pi ,\pi ).$ The periodic extension of the saw-toothed function f(t)=t on (-\pi ,\pi ).

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