7.2E: Orthogonal Sets of Vectors Exercises

Use the dot product in \(\mathbb{R}^{n}\) unless otherwise instructed.

Exercise \(\PageIndex{1}\) In each case, verify that \(B\) is an orthogonal basis of \(V\) with the given inner product and use the expansion theorem to express \(\mathbf{v}\) as a linear combination of the basis vectors.

a. \(\mathbf{v}=\left[\begin{array}{l}a \\ b\end{array}\right], B=\left\{\left[\begin{array}{r}1 \\ -1\end{array}\right],\left[\begin{array}{l}1 \\ 0\end{array}\right]\right\}, V=\mathbb{R}^{2}\)

\(\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^{T} A \mathbf{w}\) where \(A=\left[\begin{array}{ll}2 & 2 \\ 2 & 5\end{array}\right]\)

b. \(\mathbf{v}=\left[\begin{array}{l}a \\ b \\ c\end{array}\right], B=\left\{\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right],\left[\begin{array}{r}-1 \\ 0 \\ 1\end{array}\right],\left[\begin{array}{r}1 \\ -6 \\ 1\end{array}\right]\right\}\),

\(V=\mathbb{R}^{3},\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^{T} A \mathbf{w}\) where \(A=\left[\begin{array}{ccc}2 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2\end{array}\right]\)

c. \(\mathbf{v}=a+b x+c x^{2}, B=\left\{1 x, 2-3 x^{2}\right\}, V=\mathbf{P}_{2}\),

\(\langle p, q\rangle=p(0) q(0)+p(1) q(1)+p(-1) q(-1)\)

d. \(\mathbf{v}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\)

\(B=\left\{\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right],\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right],\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]\right\}\), \(V=\mathbf{M}_{22},\langle X, Y\rangle=\operatorname{tr}\left(X Y^{T}\right)\)

Exercise \(\PageIndex{2}\) Let \(\mathbb{R}^{3}\) have the inner product \(\left\langle(x, y, z),\left(x^{\prime}, y^{\prime}, z^{\prime}\right)\right\rangle=2 x x^{\prime}+y y^{\prime}+3 z z^{\prime}\). In each case, use the Gram-Schmidt algorithm to transform \(B\) into an orthogonal basis.
a. \(B=\{(1,1,0),(1,0,1),(0,1,1)\}\)
b. \(B=\{(1,1,1),(1,-1,1),(1,1,0)\}\)

Exercise \(\PageIndex{3}\) Let \(\mathbf{M}_{22}\) have the inner product \(\langle X, Y\rangle=\operatorname{tr}\left(X Y^{T}\right)\). In each case, use the Gram-Schmidt algorithm to transform \(B\) into an orthogonal basis.
a. \(B=\left\{\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\right\}\)
b. \(B=\left\{\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]\right\}\)

Exercise \(\PageIndex{4}\) In each case, use the Gram-Schmidt process to convert the basis \(B=\left\{1, x, x^{2}\right\}\) into an orthogonal basis of \(\mathbf{P}_{2}\).
a. \(\langle p, q\rangle=p(0) q(0)+p(1) q(1)+p(2) q(2)\)
b. \(\langle p, q\rangle=\int_{0}^{2} p(x) q(x) d x\)

Exercise \(\PageIndex{5}\) Show that \(\left\{1, x-\frac{1}{2}, x^{2}-x+\frac{1}{6}\right\}\), is an orthogonal basis of \(\mathbf{P}_{2}\) with the inner product

\[
\langle p, q\rangle=\int_{0}^{1} p(x) q(x) d x
\]

and find the corresponding orthonormal basis.

Exercise \(\PageIndex{6}\) In each case find \(U^{\perp}\) and compute \(\operatorname{dim} U\) and \(\operatorname{dim} U^{\perp}\).
a. \(U=\operatorname{span}\{(1,1,2,0),(3,-1,2,1)\), \((1,-3,-2,1)\}\) in \(\mathbb{R}^{4}\)
b. \(U=\operatorname{span}\{(1,1,0,0)\}\) in \(\mathbb{R}^{4}\)
c. \(U=\operatorname{span}\{1, x\}\) in \(\mathbf{P}_{2}\) with \(\langle p, q\rangle=p(0) q(0)+p(1) q(1)+p(2) q(2)\)
d. \(U=\operatorname{span}\{x\}\) in \(\mathbf{P}_{2}\) with \(\langle p, q\rangle=\int_{0}^{1} p(x) q(x) d x\)
e. \(U=\operatorname{span}\left\{\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]\right\}\) in \(\mathbf{M}_{22}\) with \(\langle X, Y\rangle=\operatorname{tr}\left(X Y^{T}\right)\)
f. \(U=\operatorname{span}\left\{\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\right\}\) in \(\mathbf{M}_{22}\) with \(\langle X, Y\rangle=\operatorname{tr}\left(X Y^{T}\right)\)

Exercise \(\PageIndex{7}\) Let \(\langle X, Y\rangle=\operatorname{tr}\left(X Y^{T}\right)\) in \(\mathbf{M}_{22}\). In each case find the matrix in \(U\) closest to \(A\).
a. \(U=\operatorname{span}\left\{\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\right\}\) \(A=\left[\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right]\)
b. \(U=\operatorname{span}\left\{\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right],\left[\begin{array}{rr}1 & 1 \\ 1 & -1\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]\right\}\), \(A=\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right]\)

Exercise \(\PageIndex{8}\) In \(\mathbf{P}_{2}\), let

\[
\langle p(x), q(x)\rangle=p(0) q(0)+p(1) q(1)+p(2) q(2)
\]

In each case find the polynomial in \(U\) closest to \(f(x)\).
a. \(U=\operatorname{span}\left\{1+x, x^{2}\right\}, f(x)=1+x^{2}\)
b. \(U=\operatorname{span}\left\{1,1+x^{2}\right\} ; f(x)=x\)

Exercise \(\PageIndex{9}\) Using the inner product given by \(\langle p, q\rangle=\int_{0}^{1} p(x) q(x) d x\) on \(\mathbf{P}_{2}\), write \(\mathbf{v}\) as the sum of a vector in \(U\) and a vector in \(U^{\perp}\).
a. \(\mathbf{v}=x^{2}, U=\operatorname{span}\{x+1,9 x-5\}\)
b. \(\mathbf{v}=x^{2}+1, U=\operatorname{span}\{1,2 x-1\}\)

Exercise \(\PageIndex{10}\)
a. Show that \(\{\mathbf{u}, \mathbf{v}\}\) is orthogonal if and only if \(\|\mathbf{u}+\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}\)

b. If \(\mathbf{u}=\mathbf{v}=(1,1)\) and \(\mathbf{w}=(-1,0)\), show that \(\|\mathbf{u}+\mathbf{v}+\mathbf{w}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}+\|\mathbf{w}\|^{2}\) but \(\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}\) is not orthogonal. Hence the converse to Pythagoras' theorem need not hold for more than two vectors.

Exercise \(\PageIndex{11}\) Let \(\mathbf{v}\) and \(\mathbf{w}\) be vectors in an inner product space \(V\). Show that:

a. \(\mathbf{v}\) is orthogonal to \(\mathbf{w}\) if and only if

\[
\|\mathbf{v}+\mathbf{w}\|=\|\mathbf{v}-\mathbf{w}\|
\]

b. \(\mathbf{v}+\mathbf{w}\) and \(\mathbf{v}-\mathbf{w}\) are orthogonal if and only if \(\|\mathbf{v}\|=\|\mathbf{w}\|\)

Exercise \(\PageIndex{12}\) Let \(U\) and \(W\) be subspaces of an \(n\) dimensional inner product space \(V\). Suppose \(\langle\mathbf{u}, \mathbf{v}\rangle=0\) for all \(\mathbf{u} \in U\) and \(\mathbf{w} \in W\) and \(\operatorname{dim} U+\operatorname{dim} W=n\). Show that \(U^{\perp}=W\).

Exercise \(\PageIndex{13}\) If \(U\) and \(W\) are subspaces of an inner product space, show that \((U+W)^{\perp}=U^{\perp} \cap W^{\perp}\).

Exercise \(\PageIndex{14}\)If \(X\) is any set of vectors in an inner product space \(V\), define

\[
X^{\perp}=\{\mathbf{v} \mid \mathbf{v} \text { in } V,\langle\mathbf{v}, \mathbf{x}\rangle=0 \text { for all } \mathbf{x} \text { in } X\}
\]

a. Show that \(X^{\perp}\) is a subspace of \(V\).
b. If \(U=\operatorname{span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{m}\right\}\), show that \(U^{\perp}=\left\{\mathbf{u}_{1}, \ldots, \mathbf{u}_{m}\right\}^{\perp}\).

c. If \(X \subseteq Y\), show that \(Y^{\perp} \subseteq X^{\perp}\).

d. Show that \(X^{\perp} \cap Y^{\perp}=(X \cup Y)^{\perp}\).

Exercise \(\PageIndex{15}\)If \(\operatorname{dim} V=n\) and \(\mathbf{w} \neq \mathbf{0}\) in \(V\), show that \(\operatorname{dim}\{\mathbf{v} \mid \mathbf{v}\) in \(V,\langle\mathbf{v}, \mathbf{w}\rangle=0\}=n-1\).

Exercise \(\PageIndex{16}\) If the Gram-Schmidt process is used on an orthogonal basis \(\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\}\) of \(V\), show that \(\mathbf{f}_{k}=\mathbf{v}_{k}\) holds for each \(k=1,2, \ldots, n\). That is, show that the algorithm reproduces the same basis.

Exercise \(\PageIndex{17}\) If \(\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \ldots, \mathbf{f}_{n-1}\right\}\) is orthonormal in an inner product space of dimension \(n\), prove that there are exactly two vectors \(\mathbf{f}_{n}\) such that \(\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \ldots, \mathbf{f}_{n-1}, \mathbf{f}_{n}\right\}\) is an orthonormal basis.

Exercise \(\PageIndex{18}\) Let \(U\) be a finite dimensional subspace of an inner product space \(V\), and let \(\mathbf{v}\) be a vector in \(V\).

a. Show that \(\mathbf{v}\) lies in \(U\) if and only if \(\mathbf{v}=\operatorname{proj}_{U}(\mathbf{v})\).

b. If \(V=\mathbb{R}^{3}\), show that \((-5,4,-3)\) lies in \(\operatorname{span}\{(3,-2,5),(-1,1,1)\}\) but that \((-1,0,2)\) does not.

Exercise \(\PageIndex{19}\) Let \(\mathbf{n} \neq \mathbf{0}\) and \(\mathbf{w} \neq \mathbf{0}\) be nonparallel vectors in \(\mathbb{R}^{3}\) (as in Chapter 4).

a. Show that \(\left\{\mathbf{n}, \mathbf{n} \times \mathbf{w}, \mathbf{w}-\frac{\mathbf{n} \cdot \mathbf{w}}{\|\mathbf{n}\|^{2}} \mathbf{n}\right\}\) is an orthogonal basis of \(\mathbb{R}^{3}\).

b. Show that span \(\left\{\mathbf{n} \times \mathbf{w}, \mathbf{w}-\frac{\mathbf{n} \cdot \mathbf{w}}{\|\mathbf{n}\|^{2}} \mathbf{n}\right\}\) is the plane through the origin with normal \(\mathbf{n}\).

Exercise \(\PageIndex{20}\) Let \(E=\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \ldots, \mathbf{f}_{n}\right\}\) be an orthonormal basis of \(V\).

a. Show that \(\langle\mathbf{v}, \mathbf{w}\rangle=C_{E}(\mathbf{v}) \cdot C_{E}(\mathbf{w})\) for all \(\langle\mathbf{v}, \mathbf{w}\rangle\) in \(V\).

b. If \(P=\left[p_{i j}\right]\) is an \(n \times n\) matrix, define \(\mathbf{b}_{i}=p_{i 1} \mathbf{f}_{1}+\cdots+p_{i n} \mathbf{f}_{n}\) for each \(i\). Show that \(B=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \ldots, \mathbf{b}_{n}\right\}\) is an orthonormal basis if and only if \(P\) is an orthogonal matrix.

Exercise \(\PageIndex{21}\) Let \(\left\{\mathbf{f}_{1}, \ldots, \mathbf{f}_{n}\right\}\) be an orthogonal basis of \(V\). If \(\mathbf{v}\) and \(\mathbf{w}\) are in \(V\), show that

\[
\langle\mathbf{v}, \mathbf{w}\rangle=\frac{\left\langle\mathbf{v}, \mathbf{f}_{1}\right\rangle\left\langle\mathbf{w}, \mathbf{f}_{1}\right\rangle}{\left\|\mathbf{f}_{1}\right\|^{2}}+\cdots+\frac{\left\langle\mathbf{v}, \mathbf{f}_{n}\right\rangle\left\langle\mathbf{w}, \mathbf{f}_{n}\right\rangle}{\left\|\mathbf{f}_{n}\right\|^{2}}
\]

Exercise \(\PageIndex{22}\) Let \(\left\{\mathbf{f}_{1}, \ldots, \mathbf{f}_{n}\right\}\) be an orthonormal basis of \(V\), and let \(\mathbf{v}=v_{1} \mathbf{f}_{1}+\cdots+v_{n} \mathbf{f}_{n}\) and \(\mathbf{w}=w_{1} \mathbf{f}_{1}+\cdots+w_{n} \mathbf{f}_{n}\). Show that

\[
\langle\mathbf{v}, \mathbf{w}\rangle=v_{1} w_{1}+\cdots+v_{n} w_{n}
\]

and

\[
\|\mathbf{v}\|^{2}=v_{1}^{2}+\cdots+v_{n}^{2}
\]

\section{(Parseval's formula).}

Exercise \(\PageIndex{23}\) Let \(\mathbf{v}\) be a vector in an inner product space \(V\).

a. Show that \(\|\mathbf{v}\| \geq\left\|\operatorname{proj}_{U} \mathbf{v}\right\|\) holds for all finite dimensional subspaces \(U\). [Hint: Pythagoras' theorem.]

b. If \(\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \ldots, \mathbf{f}_{m}\right\}\) is any orthogonal set in \(V\), prove Bessel's inequality:

\[
\frac{\left\langle\mathbf{v}, \mathbf{f}_{1}\right\rangle^{2}}{\left\|\mathbf{f}_{1}\right\|^{2}}+\cdots+\frac{\left\langle\mathbf{v}, \mathbf{f}_{m}\right\rangle^{2}}{\left\|\mathbf{f}_{m}\right\|^{2}} \leq\|\mathbf{v}\|^{2}
\]

Exercise \(\PageIndex{24}\) Let \(B=\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \ldots, \mathbf{f}_{n}\right\}\) be an orthogonal basis of an inner product space \(V\). Given \(\mathbf{v} \in V\), let \(\theta_{i}\) be the angle between \(\mathbf{v}\) and \(\mathbf{f}_{i}\) for each \(i\) (see Exercise 10.1.31). Show that

\[
\cos ^{2} \theta_{1}+\cos ^{2} \theta_{2}+\cdots+\cos ^{2} \theta_{n}=1
\]

[The \(\cos \theta_{i}\) are called direction cosines for \(\mathbf{v}\) corresponding to \(B\).]

Exercise \(\PageIndex{25}\)
a. Let \(S\) denote a set of vectors in a finite dimensional inner product space \(V\), and suppose that \(\langle\mathbf{u}, \mathbf{v}\rangle=0\) for all \(\mathbf{u}\) in \(S\) implies \(\mathbf{v}=\mathbf{0}\). Show that \(V=\operatorname{span} S\). [Hint: Write \(U=\operatorname{span} S\) and use Theorem 10.2.6.]

b. Let \(A_{1}, A_{2}, \ldots, A_{k}\) be \(n \times n\) matrices. Show that the following are equivalent.

i. If \(A_{i} \mathbf{b}=\mathbf{0}\) for all \(i\) (where \(\mathbf{b}\) is a column in \(\left.\mathbb{R}^{n}\right)\), then \(\mathbf{b}=\mathbf{0}\).

ii. The set of all rows of the matrices \(A_{i}\) spans \(\mathbb{R}^{n}\).

Exercise \(\PageIndex{26}\)Let \(\left[x_{i}\right)=\left(x_{1}, x_{2}, \ldots\right)\) denote a sequence of real numbers \(x_{i}\), and let

\[
V=\left\{\left[x_{i}\right) \mid \text { only finitely many } x_{i} \neq 0\right\}
\]

Define componentwise addition and scalar multiplication on \(V\) as follows:

\[
\left[x_{i}\right)+\left[y_{i}\right)=\left[x_{i}+y_{i}\right) \text {, and } a\left[x_{i}\right)=\left[a x_{i}\right) \text { for } a \text { in } \mathbb{R} .
\]

Given \(\left[x_{i}\right)\) and \(\left[y_{i}\right)\) in \(V\), define \(\left\langle\left[x_{i}\right),\left[y_{i}\right)\right\rangle=\sum_{i=0}^{\infty} x_{i} y_{i}\). (Note that this makes sense since only finitely many \(x_{i}\) and \(y_{i}\) are nonzero.) Finally define

\[
U=\left\{\left[x_{i}\right) \text { in } V \mid \sum_{i=0}^{\infty} x_{i}=0\right\}
\]

a. Show that \(V\) is a vector space and that \(U\) is a subspace.

b. Show that \(\langle\),\(\rangle is an inner product on V\).

c. Show that \(U^{\perp}=\{\mathbf{0}\}\).

d. Hence show that \(U \oplus U^{\perp} \neq V\) and \(U \neq U^{\perp \perp}\).