Chapter Questions
Let $\mathbb{R}^2$ have the standard dot product. Classify the following pairs of vectors as(i) basis, (ii) orthogonal basis, and/or (iii) orthonormal basis:(a) $\mathbf{v}_1=\left(\begin{array}{r}-1 \\ 2\end{array}\right), \mathbf{v}_2=\left(\begin{array}{l}2 \\ 1\end{array}\right)$;(b) $\mathbf{v}_1=\left(\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}-\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right)$;(c) $\mathbf{v}_1=\left(\begin{array}{l}-1 \\ -1\end{array}\right), \mathbf{v}_2=\left(\begin{array}{l}2 \\ 2\end{array}\right) ;$(d) $\mathbf{v}_1=\left(\begin{array}{l}2 \\ 3\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}1 \\ -6\end{array}\right)$;(e) $\mathbf{v}_1=\left(\begin{array}{r}-1 \\ 0\end{array}\right), \mathbf{v}_2=\left(\begin{array}{l}0 \\ 3\end{array}\right)$;(f) $\mathbf{v}_1=\left(\begin{array}{c}\frac{3}{5} \\ \frac{4}{5}\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}-\frac{4}{5} \\ \frac{3}{5}\end{array}\right)$.
Use the Gram-Schmidt process to determine an orthonormal basis for $\mathbb{R}^3$ starting with the following sets of vectors:(a) $\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 2 \\ 1\end{array}\right)$;(b) $\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}0 \\ 1 \\ -1\end{array}\right),\left(\begin{array}{r}1 \\ 0 \\ -1\end{array}\right)$;(c) $\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{l}4 \\ 5 \\ 0\end{array}\right),\left(\begin{array}{r}2 \\ 3 \\ -1\end{array}\right)$.
Determine which of the following matrices are $(i)$ orthogonal; (ii) proper orthogonal.(a) $\left(\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right)$,(b)$$\left(\begin{array}{rr}\frac{12}{13} & \frac{5}{13} \\-\frac{5}{13} & \frac{12}{13}\end{array}\right)$$(c) $\left(\begin{array}{rrr}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1\end{array}\right)$,(d)(d) $\left(\begin{array}{rrr}-\frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ \frac{2}{3} & -\frac{1}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{2}{3} & -\frac{1}{3}\end{array}\right)$,(e) $\left(\begin{array}{lll}\frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6}\end{array}\right)$$\left(\begin{array}{rrr}\frac{3}{5} & 0 & \frac{4}{5} \\ -\frac{4}{13} & \frac{12}{13} & \frac{3}{13} \\ -\frac{48}{65} & -\frac{5}{13} & \frac{36}{65}\end{array}\right)$,$(g)$$\left(\begin{array}{rrr}\frac{2}{3} & -\frac{\sqrt{2}}{6} & \frac{\sqrt{2}}{2} \\ -\frac{2}{3} & \frac{\sqrt{2}}{6} & \frac{\sqrt{2}}{2} \\ \frac{1}{3} & \frac{2 \sqrt{2}}{3} & 0\end{array}\right)$.
Determine which of the vectors $\mathbf{v}_1=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}-2 \\ 2 \\ 2\end{array}\right), \mathbf{v}_3=\left(\begin{array}{r}2 \\ -1 \\ -3\end{array}\right), \mathbf{v}_4=\left(\begin{array}{r}-1 \\ 3 \\ 4\end{array}\right)$, is orthogonal to (a) the line spanned by $\left(\begin{array}{r}1 \\ 3 \\ -2\end{array}\right) ;(b)$ the plane spanned by $\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 1\end{array}\right)$; (c) the plane defined by $x-y-z=0 ;(d)$ the kernel of the matrix $\left(\begin{array}{lll}1 & -1 & -1 \\ 3 & -2 & -4\end{array}\right)$; (e) the image of the matrix $\left(\begin{array}{rr}-3 & 1 \\ 3 & -1 \\ -1 & 0\end{array}\right)$;(f) the cokernel of the matrix $\left(\begin{array}{rrr}-1 & 0 & 3 \\ 2 & 1 & -2 \\ 3 & 1 & -5\end{array}\right)$.
Write the following polynomials as linear combinations of monic Legendre polynomials.Use orthogonality to compute the coefficients:(a) $t^3$(b) $t^4+t^2$(c) $7 t^4+2 t^3-t$.
Let $\mathbb{R}^3$ have the standard dot product. Classify the following sets of vectors as(i) basis, (ii) orthogonal basis, and/or (iii) orthonormal basis:(a) $\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$;(b) $\left(\begin{array}{c}-\frac{4}{13} \\ \frac{3}{5} \\ -\frac{48}{65}\end{array}\right),\left(\begin{array}{c}\frac{12}{13} \\ 0 \\ -\frac{5}{13}\end{array}\right),\left(\begin{array}{c}\frac{3}{13} \\ \frac{4}{5} \\ \frac{36}{65}\end{array}\right)$;(c) $\left(\begin{array}{r}0 \\ \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}}\end{array}\right),\left(\begin{array}{r}-\frac{1}{\sqrt{2}} \\ 0 \\ \frac{1}{\sqrt{2}}\end{array}\right),\left(\begin{array}{r}\frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ 0\end{array}\right)$.
Use the Gram-Schmidt process to construct an orthonormal basis for $\mathbb{R}^4$ starting with the following sets of vectors: (a) $(1,0,1,0)^T,(0,1,0,-1)^T,(1,0,0,1)^T,(1,1,1,1)^T$; (b) $(1,0,0,1)^T,(4,1,0,0)^T,(1,0,2,1)^T,(0,2,0,1)^T$.
(a) Show that $R=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right)$, a reflection matrix, and $Q=\left(\begin{array}{ccc}\cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right)$, representing a rotation by the angle $\theta$ around the $z$-axis, are both orthogonal. (b) Verify that the products $R Q$ and $Q R$ are also orthogonal. (c) Which of the preceding matrices, $R, Q, R Q, Q R$, are proper orthogonal?
Find the orthogonal projection of the vector $\mathbf{v}=(1,1,1)^T$ onto the following subspaces, using the indicated orthonormal/orthogonal bases: (a) the line in the direction $\left(-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)^T$(b) the line spanned by $(2,-1,3)^T$;(c) the plane spanned by $(1,1,0)^T,(-2,2,1)^T$;(d) the plane spanned by $\left(-\frac{3}{5}, \frac{4}{5}, 0\right)^T,\left(\frac{4}{13}, \frac{3}{13},-\frac{12}{13}\right)^T$.
(a) Find the monic Legendre polynomial of degree 5 using the Gram-Schmidt process. Check your answer using the Rodrigues formula. (b) Use orthogonality to write $t^5$ as a linear combination of Legendre polynomials. (c) Repeat the exercise for degree 6.
Repeat Exercise 4.1.1, but use the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+\frac{1}{9} v_2 w_2$ instead of the dot product.
Try the Gram-Schmidt procedure on the vectors $\left(\begin{array}{r}1 \\ -1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}0 \\ -1 \\ 1 \\ 2\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}2 \\ 2 \\ -2 \\ 1\end{array}\right)$. What happens? Can you explain why you are unable to complete the algorithm?
True or false: (a) If $Q$ is an improper $2 \times 2$ orthogonal matrix, then $Q^2=\mathrm{I}$.(b) If $Q$ is an improper $3 \times 3$ orthogonal matrix, then $Q^2=\mathrm{I}$.
Find the orthogonal projection of $\mathbf{v}=(1,2,-1,2)^T$ onto the following subspaces:(a) the span of $\left(\begin{array}{r}1 \\ -1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ 1 \\ 0 \\ -1\end{array}\right)$;(b) the image of the matrix $\left(\begin{array}{rr}1 & 2 \\ -1 & 1 \\ 0 & 3 \\ -1 & 1\end{array}\right)$;(c) the kernel of the matrix $\left(\begin{array}{rrrr}1 & -1 & 0 & 1 \\ -2 & 1 & 1 & 0\end{array}\right)$;(d) the subspace orthogonal to $\mathbf{a}=(1,-1,0,1)^T$. Warning. Make sure you have an orthogonal basis before applying formula (4.42)!
(a) Explain why $q_n$ is the unique monic polynomial that satisfies (4.55). (b) Use this characterization to directly construct $q_5(t)$.
Show that the standard basis vectors $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ form an orthogonal basis with respect to the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+2 v_2 w_2+3 v_3 w_3$ on $\mathbb{R}^3$. Find an orthonormal basis for this inner product space.
Use the Gram-Schmidt process to construct an orthonormal basis for the following subspaces of $\mathbb{R}^3$ : (a) the plane spanned by $(0,2,1)^T,(1,-2,-1)^T ;$ (b) the plane defined by the equation $2 x-y+3 z=0 ;(c)$ the set of all vectors orthogonal to $(1,-1,-2)^T$.
(a) Prove that, for all $\theta, \varphi, \psi$,$$Q=\left(\begin{array}{ccc}\cos \varphi \cos \psi-\cos \theta \sin \varphi \sin \psi & \sin \varphi \cos \psi+\cos \theta \cos \varphi \sin \psi & \sin \theta \sin \psi \\-\cos \varphi \sin \psi-\cos \theta \sin \varphi \cos \psi & -\sin \varphi \sin \psi+\cos \theta \cos \varphi \cos \psi & \sin \theta \cos \psi \\\sin \theta \sin \varphi & -\sin \theta \cos \varphi & \cos \theta\end{array}\right)$$is a proper orthogonal matrix. (b) Write down a formula for $Q^{-1}$.Remark. It can be shown that every proper orthogonal matrix can be parameterized in this manner; $\theta, \varphi, \psi$ are known as the Euler angles, and play an important role in applications in mechanics and geometry, $[31 ;$ p. 147].
Find the orthogonal projection of the vector $\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$ onto the image of $\left(\begin{array}{rr}3 & 2 \\ 2 & -2 \\ 1 & -2\end{array}\right)$.
Prove that the even (odd) degree Legendre polynomials are even (odd) functions of $t$.
Find all values of $a$ such that the vectors $\left(\begin{array}{l}a \\ 1\end{array}\right),\left(\begin{array}{r}-a \\ 1\end{array}\right)$ form an orthogonal basis of $\mathbb{R}^2$ under (a) the dot product; (b) the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=3 v_1 w_1+2 v_2 w_2$; (c) the inner product prescribed by the positive definite matrix $K=\left(\begin{array}{rr}2 & -1 \\ -1 & 3\end{array}\right)$.
Find an orthogonal basis of the subspace spanned by the vectors $\mathbf{w}_1=(1,-1,-1,1,1)^T$, $\mathbf{w}_2=(2,1,4,-4,2)^T$, and $\mathbf{w}_3=(5,-4,-3,7,1)^T$.
(a) Show that if $y_1^2+y_2^2+y_3^2+y_4^2=1$, then the matrix$$Q=\left(\begin{array}{ccc}y_1^2+y_2^2-y_3^2-y_4^2 & 2\left(y_2 y_3+y_1 y_4\right) & 2\left(y_2 y_4-y_1 y_3\right) \\2\left(y_2 y_3-y_1 y_4\right) & y_1^2-y_2^2+y_3^2-y_4^2 & 2\left(y_3 y_4+y_1 y_2\right) \\2\left(y_2 y_4+y_1 y_3\right) & 2\left(y_3 y_4-y_1 y_2\right) & y_1^2-y_2^2-y_3^2+y_4^2\end{array}\right)$$is a proper orthogonal matrix. The numbers $y_1, y_2, y_3, y_4$ are known as Cayley-Klein parameters. (b) Write down a formula for $Q^{-1}$. (c) Prove the formulas$$y_1=\cos \frac{\varphi+\psi}{2} \cos \frac{\theta}{2}, y_2=\cos \frac{\varphi-\psi}{2} \sin \frac{\theta}{2}, y_3=\sin \frac{\varphi-\psi}{2} \sin \frac{\theta}{2}, y_4=\sin \frac{\varphi+\psi}{2} \cos \frac{\theta}{2} \text {, }$$relating the Cayley-Klein parameters and the Euler angles of Exercise 4.3.4, cf. [31; $\$ 4-5]$.
Find the orthogonal projection of the vector $\mathbf{v}=(1,3,-1)^T$ onto the plane spanned by $(-1,2,1)^T,(2,1,-3)^T$ by first using the Gram-Schmidt process to construct an orthogonal basis.
Prove that if $p(t)=p(-t)$ is an even polynomial, then all the odd-order coefficients $c_{2 j+1}=0$ in its Legendre expansion (4.56) vanish.
Find all possible values of $a$ and $b$ in the inner product $\langle\mathbf{v}, \mathbf{w}\rangle=a v_1 w_1+b v_2 w_2$ that make the vectors $(1,2)^T,(-1,1)^T$, an orthogonal basis in $\mathbb{R}^2$.
Find an orthonormal basis for the following subspaces of $\mathbb{R}^4$ : (a) the span of the vectors $\left(\begin{array}{r}1 \\ 1 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 0 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 2 \\ 1\end{array}\right)$(b) the kernel of the matrix $\left(\begin{array}{rrrr}2 & 1 & 0 & -1 \\ 3 & 2 & -1 & -1\end{array}\right)$;(c) the coimage of the preceding matrix;(d) the image of the matrix $\left(\begin{array}{rrr}1 & -2 & 2 \\ 2 & -4 & 1 \\ 0 & 0 & -1 \\ -2 & 4 & 5\end{array}\right)$;(e) the cokernel of the preceding matrix; $(f)$ the set of all vectors orthogonal to $(1,1,-1,-1)^T$.
(a) Prove that the transpose of an orthogonal matrix is also orthogonal. (b) Explain why the rows of an $n \times n$ orthogonal matrix also form an orthonormal basis of $\mathbb{R}^n$.
Find the orthogonal projection of $\mathbf{v}=(1,2,-1,2)^T$ onto the span of $(1,-1,2,5)^T$ and $(2,1,0,-1)^T$ using the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=4 v_1 w_1+3 v_2 w_2+2 v_3 w_3+v_4 w_4$.
Write out an explicit Rodrigues-type formula for the monic Legendre polynomial $q_k(t)$ and its norm.
Answer Exercise 4.1.6 for the vectors (a) $(2,3)^T,(-2,2)^T$;(b) $(1,4)^T,(2,1)^T$.
Find orthonormal bases for the four fundamental subspaces associated with the following matrices:(a) $\left(\begin{array}{rr}1 & -1 \\ -3 & 3\end{array}\right)$,(b) $\left(\begin{array}{rrr}-1 & 0 & 2 \\ 1 & 1 & -1 \\ 0 & 1 & 1\end{array}\right)$,(c) $\left(\begin{array}{rrrr}1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 \\ -1 & 2 & 0 & 1\end{array}\right)$,(d) $\left(\begin{array}{rrr}1 & 2 & 1 \\ 0 & -2 & 1 \\ -1 & 0 & -2 \\ 1 & -2 & 3\end{array}\right)$
Prove that the inverse of an orthogonal matrix is orthogonal.
Redo Exercise 4.4 .2 using(i) the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=2 v_1 w_1+2 v_2 w_2+v_3 w_3$;(ii) the inner product induced by the positive definite matrix $K=\left(\begin{array}{rrr}2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right)$.
Write out an explicit Rodrigues-type formula for an orthonormal basis $Q_0(t), \ldots, Q_n(t)$ for the space of polynomials of degree $\leq n$ under the inner product (4.52).
Find an inner product such that the vectors $(-1,2)^T$ and $(1,2)^T$ form an orthonormal basis of $\mathbb{R}^2$.
Construct an orthonormal basis of $\mathbb{R}^2$ for the nonstandard inner products(a) $\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T\left(\begin{array}{ll}3 & 0 \\ 0 & 5\end{array}\right) \mathbf{y}$,(b) $\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T\left(\begin{array}{rr}4 & -1 \\ -1 & 1\end{array}\right) \mathbf{y}$,(c) $\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T\left(\begin{array}{ll}2 & 1 \\ 1 & 3\end{array}\right) \mathbf{y}$.
Show that if $Q$ is a proper orthogonal matrix, and $R$ is obtained from $Q$ by interchanging two rows, then $R$ is an improper orthogonal matrix.
(a) Prove that the set of all vectors orthogonal to a given subspace $V \subset \mathbb{R}^m$ forms a subspace. (b) Find a basis for the set of all vectors in $\mathbb{R}^4$ that are orthogonal to the subspace spanned by $(1,2,0,-1)^T,(2,0,3,1)^T$.
Use the Rodrigues formula to prove (4.59).
True or false: If $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are a basis for $\mathbb{R}^3$, then they form an orthogonal basis under some appropriately weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=a v_1 w_1+b v_2 w_2+c v_3 w_3$.
Construct an orthonormal basis for $\mathbb{R}^3$ with respect to the inner products defined by the following positive definite matrices:(a) $\left(\begin{array}{rrr}4 & -2 & 0 \\ -2 & 3 & -1 \\ 0 & -1 & 2\end{array}\right)$,(b) $\left(\begin{array}{rrr}3 & -1 & 1 \\ -1 & 4 & -2 \\ 1 & -2 & 4\end{array}\right)$
Show that the product of two proper orthogonal matrices is also proper orthogonal. What can you say about the product of two improper orthogonal matrices? What about an improper times a proper orthogonal matrix?
Let $\mathbf{u}_1, \ldots, \mathbf{u}_k$ be an orthonormal basis for the subspace $W \subset \mathbb{R}^m$. Let $A=\left(\mathbf{u}_1 \mathbf{u}_2 \ldots \mathbf{u}_k\right)$ be the $m \times k$ matrix whose columns are the orthonormal basis vectors, and define $P=A A^T$ to be the corresponding projection matrix. (a) Given $\mathbf{v} \in \mathbb{R}^n$, prove that its orthogonal projection $\mathbf{w} \in W$ is given by matrix multiplication: $\mathbf{w}=P \mathbf{v}$.(b) Prove that $P=P^T$ is symmetric. (c) Prove that $P$ is idempotent: $P^2=P$. Give a geometrical explanation of this fact. (d) Prove that $\operatorname{rank} P=k$. (e) Write out the projection matrix corresponding to the subspaces spanned by(i) $\left(\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right)$,(ii)$\left(\begin{array}{r}\frac{2}{3} \\ -\frac{2}{3} \\ \frac{1}{3}\end{array}\right)$(iii)$$\left(\begin{array}{c}\frac{1}{\sqrt{6}} \\-\frac{2}{\sqrt{6}} \\\frac{1}{\sqrt{6}}\end{array}\right),\left(\begin{array}{c}\frac{1}{\sqrt{3}} \\\frac{1}{\sqrt{3}} \\\frac{1}{\sqrt{3}}\end{array}\right)$$(iv)$$\left(\begin{array}{r}\frac{1}{2} \\\frac{1}{2} \\\frac{1}{2} \\-\frac{1}{2}\end{array}\right), \quad\left(\begin{array}{r}\frac{1}{2} \\-\frac{1}{2} \\\frac{1}{2} \\\frac{1}{2}\end{array}\right), \quad\left(\begin{array}{r}\frac{1}{2} \\\frac{1}{2} \\-\frac{1}{2} \\\frac{1}{2}\end{array}\right) .$$
A proof of the formula in (4.61) for the norms of the Legendre polynomials is based on the following steps. (a) First, prove that $\left\|R_{k, k}\right\|^2=(-1)^k(2 k) ! \int_{-1}^1\left(t^2-1\right)^k d t$ by a repeated integration by parts. (b) Second, prove that $\int_{-1}^1\left(t^2-1\right)^k d t=(-1)^k \frac{2^{2 k+1}(k !)^2}{(2 k+1) !}$ by using the change of variables $t=\cos \theta$ in the integral. The resulting trigonometric integral can be done by another repeated integration by parts. (c) Finally, use the Rodrigues formula to complete the proof.
The cross product between two vectors in $\mathbb{R}^3$ is the vector defined by the formula$$\mathbf{v} \times \mathbf{w}=\left(\begin{array}{c}v_2 w_3-v_3 w_2 \\v_3 w_1-v_1 w_3 \\v_1 w_2-v_2 w_1\end{array}\right), \quad \text { where }=\left(\begin{array}{c}v_1 \\v_2 \\v_3\end{array}\right), \quad \mathbf{w}=\left(\begin{array}{c}w_1 \\w_2 \\w_3\end{array}\right) .$$(a) Show that $\mathbf{u}=\mathbf{v} \times \mathbf{w}$ is orthogonal, under the dot product, to both $\mathbf{v}$ and $\mathbf{w}$.(b) Show that $\mathbf{v} \times \mathbf{w}=\mathbf{0}$ if and only if $\mathbf{v}$ and $\mathbf{w}$ are parallel. (c) Prove that if $\mathbf{v}, \mathbf{w} \in \mathbb{R}^3$ are orthogonal nonzero vectors, then $\mathbf{u}=\mathbf{v} \times \mathbf{w}, \mathbf{v}, \mathbf{w}$ form an orthogonal basis of $\mathbb{R}^3$.(d) True or false: If $\mathbf{v}, \mathbf{w} \in \mathbb{R}^3$ are orthogonal unit vectors, then $\mathbf{v}, \mathbf{w}$ and $\mathbf{u}=\mathbf{v} \times \mathbf{w}$ form an orthonormal basis of $\mathbb{R}^3$.
Redo Exercise 4.2.1 using(i) the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=3 v_1 w_1+2 v_2 w_2+v_3 w_3$;(ii) the inner product induced by the positive definite matrix $K=\left(\begin{array}{rrr}2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right)$.
True or false: (a) A matrix whose columns form an orthogonal basis of $\mathbb{R}^n$ is an orthogonal matrix. (b) A matrix whose rows form an orthonormal basis of $\mathbb{R}^n$ is an orthogonal matrix. (c) An orthogonal matrix is symmetric if and only if it is a diagonal matrix.
Let $\mathbf{w}_1, \ldots, \mathbf{w}_n$ be an arbitrary basis of the subspace $W \subset \mathbb{R}^m$. Let $A=\left(\mathbf{w}_1, \ldots, \mathbf{w}_n\right)$ be the $m \times n$ matrix whose columns are the basis vectors, so that $W=\operatorname{img} A$ and $\operatorname{rank} A=n$. (a) Prove that the corresponding projection matrix $P=A\left(A^T A\right)^{-1} A^T$ is idempotent: $P^2=P$. (b) Prove that $P$ is symmetric. (c) Prove that $\operatorname{img} P=W$. (d) (e) Prove that the orthogonal projection of $\mathbf{v} \in \mathbb{R}^n$ onto $\mathbf{w} \in W$ is obtained by multiplying by the projection matrix: $\mathbf{w}=P \mathbf{v}$. (f) Show that if $A$ is nonsingular, then $P=\mathrm{I}$. How do you interpret this in light of part $(e)$ ? $(g)$ Explain why Exercise 4.4 .9 is a special case of this result. (h) Show that if $A=Q R$ is the factorization of $A$ given in Exercise 4.3 .32 , then $P=Q Q^T$. Why is $P \neq \mathrm{I}$ ?
(a) Find the roots, $P_n(t)=0$, of the Legendre polynomials $P_2, P_3$ and $P_4$. (b) Prove that for $0 \leq j \leq k$, the polynomial $R_{j, k}(t)$ defined in (4.62) has roots of order $k-j$ at $t= \pm 1$, and $j$ additional simple roots lying between -1 and 1 . (c) Conclude that all $k$ roots of the Legendre polynomial $P_k(t)$ are real and simple, and that they lie in the interval $-1<t<1$.
Prove that every orthonormal basis of $\mathbb{R}^2$ under the standard dot product has the form $\mathbf{u}_1=\left(\begin{array}{c}\cos \theta \\ \sin \theta\end{array}\right)$ and $\mathbf{u}_2= \pm\left(\begin{array}{r}-\sin \theta \\ \cos \theta\end{array}\right)$ for some $0 \leq \theta<2 \pi$ and some choice of \pm sign.
(a) How many orthonormal bases does $\mathbb{R}$ have? (b) What about $\mathbb{R}^2$ ? (c) Does your answer change if you use a different inner product? Justify your answers.
Write down all diagonal $n \times n$ orthogonal matrices.
Use the projection matrix method of Exercise 4.4.10 to find the orthogonal projection of $\mathbf{v}=(1,0,0,0)^T$ onto the image of the following matrices:(a) $\left(\begin{array}{r}5 \\ -5 \\ -7 \\ 1\end{array}\right)$,(b) $\left(\begin{array}{rr}1 & 0 \\ -1 & 2 \\ 0 & -1 \\ 1 & 2\end{array}\right)$,(c) $\left(\begin{array}{rr}2 & -1 \\ -3 & 1 \\ 1 & -2 \\ 1 & 2\end{array}\right)$(d) $\left(\begin{array}{rrr}0 & 1 & -1 \\ 0 & -1 & 2 \\ 1 & 1 & 1 \\ -2 & -1 & 0\end{array}\right)$
Construct polynomials $P_0, P_1, P_2$, and $P_3$ of degree $0,1,2$, and 3 , respectively, that are orthogonal with respect to the inner products (a) $\langle f, g\rangle=\int_1^2 f(t) g(t) d t,(b)\langle f, g\rangle=$ $\int_0^1 f(t) g(t) t d t, \quad(c)\langle f, g\rangle=\int_{-1}^1 f(t) g(t) t^2 d t,(d)\langle f, g\rangle=\int_{-\infty}^{\infty} f(t) g(t) e^{-|t|} d t .$.
Given angles $\theta, \varphi, \psi$, prove that the vectors $\mathbf{u}_1=\left(\begin{array}{c}\cos \psi \cos \varphi-\cos \theta \sin \varphi \sin \psi \\ -\sin \psi \cos \varphi-\cos \theta \sin \varphi \cos \psi \\ \sin \theta \sin \varphi\end{array}\right)$, $\mathbf{u}_2=\left(\begin{array}{c}\cos \psi \sin \varphi+\cos \theta \cos \varphi \sin \psi \\ -\sin \psi \sin \varphi+\cos \theta \cos \varphi \cos \psi \\ -\sin \theta \cos \varphi\end{array}\right), \mathbf{u}_3=\left(\begin{array}{c}\sin \theta \sin \varphi \\ \sin \theta \cos \varphi \\ \cos \theta\end{array}\right)$, form an orthonormal basis of $\mathbb{R}^3$ under the standard dot product. Remark. It can be proved, [31; p. 147], that every orthonormal basis of $\mathbb{R}^3$ has the form $\mathbf{u}_1, \mathbf{u}_2, \pm \mathbf{u}_3$ for some choice of angles $\theta, \varphi, \psi$.
True or false: Reordering the original basis before starting the Gram-Schmidt process leads to the same orthogonal basis.
Prove that an upper triangular matrix $U$ is orthogonal if and only if $U$ is a diagonal matrix. What are its diagonal entries?
Find the orthogonal complement $W^{\perp}$ of the subspaces $W \subset \mathbb{R}^3$ spanned by the indicated vectors. What is the dimension of $W^{\perp}$ in each case?(a) $\left(\begin{array}{r}3 \\ -1 \\ 1\end{array}\right)$,(b) $\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{l}2 \\ 0 \\ 1\end{array}\right)$,(c) $\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right),\left(\begin{array}{l}2 \\ 4 \\ 6\end{array}\right)$,(d) $\left(\begin{array}{r}0 \\ 1 \\ -1\end{array}\right),\left(\begin{array}{r}-2 \\ 3 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 2 \\ 0\end{array}\right)$,(e) $\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$.
Find the first four orthogonal polynomials on the interval $[0,1]$ for the weighted $\mathrm{L}^2$ inner product with weight $w(t)=t^2$.
(a) Show that $\mathbf{v}_1, \ldots, \mathbf{v}_n$ form an orthonormal basis of $\mathbb{R}^n$ for the inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T K \mathbf{w}$ for $K>0$ if and only if $A^T K A=\mathrm{I}$, where $A=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$.(b) Prove that every basis of $\mathbb{R}^n$ is an orthonormal basis with respect to some inner product. Is the inner product uniquely determined? (c) Find the inner product on $\mathbb{R}^2$ that makes $\mathbf{v}_1=(1,1)^T, \mathbf{v}_2=(2,3)^T$ into an orthonormal basis. (d) Find the inner product on $\mathbb{R}^3$ that makes $\mathbf{v}_1=(1,1,1)^T, \mathbf{v}_2=(1,1,2)^T, \mathbf{v}_3=(1,2,3)^T$ an orthonormal basis.
Suppose that $W \subsetneq \mathbb{R}^n$ is a proper subspace, and $\mathbf{u}_1, \ldots, \mathbf{u}_m$ forms an orthonormal basis of $W$. Prove that there exist vectors $\mathbf{u}_{m+1}, \ldots, \mathbf{u}_n \in \mathbb{R}^n \backslash W$ such that the complete collection $\mathbf{u}_1, \ldots, \mathbf{u}_n$ forms an orthonormal basis for $\mathbb{R}^n$.
(a) Show that the elementary row operation matrix corresponding to the interchange of two rows is an improper orthogonal matrix. (b) Are there any other orthogonal elementary matrices?
Find a basis for the orthogonal complement of the following subspaces of $\mathbb{R}^3$ : (a) the plane $3 x+4 y-5 z=0 ;(b)$ the line in the direction $(-2,1,3)^T ;(c)$ the image of the $\operatorname{matrix}\left(\begin{array}{rrrr}1 & 2 & -1 & 3 \\ -2 & 0 & 2 & 1 \\ -1 & 2 & 1 & 4\end{array}\right)$;(d) the cokernel of the same matrix.
Write down an orthogonal basis for vector space $\mathcal{P}^{(5)}$ of quintic polynomials under the inner product $\langle f, g\rangle=\int_{-2}^2 f(t) g(t) d t$.
Describe all orthonormal bases of $\mathbb{R}^2$ for the inner products(a) $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T\left(\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right) \mathbf{w}$;(b) $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T\left(\begin{array}{rr}1 & -1 \\ -1 & 2\end{array}\right) \mathbf{w}$.
Verify that the Gram-Schmidt formula (4.19) also produce an orthogonal basis of a complex vector space under a Hermitian inner product.
True or false: Applying an elementary row operation to an orthogonal matrix produces an orthogonal matrix.
Find a basis for the orthogonal complement of the following subspaces of $\mathbb{R}^4$ : (a) the set of solutions to $-x+3 y-2 z+w=0 ;$ (b) the subspace spanned by $(1,2,-1,3)^T$, $(-2,0,1,-2)^T,(-1,2,0,1)^T ;(c)$ the kernel of the matrix in Exercise 4.4.13c; $(d)$ the coimage of the same matrix.
Use the Gram-Schmidt process based on the $\mathrm{L}^2$ inner product on $[0,1]$ to construct a system of orthogonal polynomials of degree $\leq 4$. Verify that your polynomials are multiples of the modified Legendre polynomials found in Example 4.56.
Let $\mathbf{v}$ and $\mathbf{w}$ be elements of an inner product space. Prove that $\|\mathbf{v}+\mathbf{w}\|^2=\|\mathbf{v}\|^2+\|\mathbf{w}\|^2$ if and only if $\mathbf{v}, \mathbf{w}$ are orthogonal. Explain why this formula can be viewed as the generalization of the Pythagorean Theorem.
(a) Apply the complex Gram-Schmidt algorithm from Exercise 4.2 .14 to produce an orthonormal basis starting with the vectors $(1+\mathrm{i}, 1-\mathrm{i})^T,(1-2 \mathrm{i}, 5 \mathrm{i})^T \in \mathbb{C}^2$.(b) Do the same for $(1+\mathrm{i}, 1-\mathrm{i}, 2-\mathrm{i})^T,(1+2 \mathrm{i},-2 \mathrm{i}, 2-\mathrm{i})^T,(1,1-2 \mathrm{i}, \mathrm{i})^T \in \mathbb{C}^3$.
(a) Prove that every permutation matrix is orthogonal. (b) How many permutation matrices of a given size are proper orthogonal?
Decompose each of the following vectors with respect to the indicated subspace as $\mathbf{v}=\mathbf{w}+\mathbf{z}$, where $\mathbf{w} \in W, \mathbf{z} \in W^{\perp}$. (a) $\mathbf{v}=\left(\begin{array}{l}1 \\ 2\end{array}\right), W=\operatorname{span}\left\{\left(\begin{array}{r}-3 \\ 1\end{array}\right)\right\}$(b) $\mathbf{v}=\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right), W=\operatorname{span}\left\{\left(\begin{array}{r}-3 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 0 \\ 5\end{array}\right)\right\} ; \quad(c) \mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right), W=\operatorname{ker}\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 0 & 2\end{array}\right)$;(d) $\mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right), W=\operatorname{img}\left(\begin{array}{rrr}1 & 0 & 1 \\ -2 & -1 & 0 \\ 1 & 3 & -5\end{array}\right) ;(e) \mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right), W=\operatorname{ker}\left(\begin{array}{rrrr}1 & 0 & 0 & 2 \\ -2 & -1 & 1 & -3\end{array}\right)$.
Find the first four orthogonal polynomials under the Sobolev $H^1$ inner product$$\langle f, g\rangle=\int_{-1}^1\left[f(t) g(t)+f^{\prime}(t) g^{\prime}(t)\right] d t ; \text { cf. Exercise 3.1.27. }$$
Prove that if $\mathbf{v}_1, \mathbf{v}_2$ form a basis of an inner product space $V$ and $\left\|\mathbf{v}_1\right\|=\left\|\mathbf{v}_2\right\|$, then $\mathbf{v}_1+\mathbf{v}_2$ and $\mathbf{v}_1-\mathbf{v}_2$ form an orthogonal basis of $V$.
Use the complex Gram-Schmidt algorithm from Exercise 4.2.14 to construct orthonormal bases for (a) the subspace spanned by $(1-\mathrm{i}, 1,0)^T,(0,3-\mathrm{i}, 2 \mathrm{i})^T$;(b) the set of solutions to $(2-\mathrm{i}) x-2 \mathrm{i} y+(1-2 \mathrm{i}) z=0$;(c) the subspace spanned by $(-\mathrm{i}, 1,-1, \mathrm{i})^T,(0,2 \mathrm{i}, 1-\mathrm{i},-1+\mathrm{i})^T,(1, \mathrm{i},-\mathrm{i}, 1-2 \mathrm{i})^T$.
(a) Prove that if $Q$ is an orthogonal matrix, then $\|Q \mathbf{x}\|=\|\mathbf{x}\|$ for every vector $\mathbf{x} \in \mathbb{R}^n$, where $\|\cdot\|$ denotes the standard Euclidean norm. (b) Prove the converse: if $\|Q \mathbf{x}\|=\|\mathbf{x}\|$ for all $\mathrm{x} \in \mathbb{R}^n$, then $Q$ is an orthogonal matrix.
Redo Exercise 4.4.12 using the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+2 v_2 w_2+3 v_3 w_3$ instead of the dot product.
Prove the formula for $\left\|\bar{P}_k\right\|$ in (4.73) .
Suppose $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are nonzero mutually orthogonal elements of an inner product space $V$. Write down their Gram matrix. Why is it nonsingular?
Use the modified Gram-Schmidt process $(4.26-27)$ to produce orthonormal bases for the spaces spanned by the following vectors:(a) $\left(\begin{array}{r}-1 \\ 1 \\ 2\end{array}\right),\left(\begin{array}{r}-1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 3\end{array}\right)$,(b) $\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 0\end{array}\right)$,(c) $\left(\begin{array}{r}1 \\ 1 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 0 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 2 \\ 1\end{array}\right)$,(d) $\left(\begin{array}{l}2 \\ 1 \\ 3 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}0 \\ -1 \\ 2 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ 2 \\ -1 \\ 0 \\ 1\end{array}\right)$,(e) $\left(\begin{array}{l}0 \\ 1 \\ 0 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}1 \\ -1 \\ 0 \\ 1 \\ -1\end{array}\right),\left(\begin{array}{r}1 \\ 0 \\ -1 \\ 0 \\ 1\end{array}\right)$.
Show that if $A^T=-A$ is any skew-symmetric matrix, then its Cayley Transform $Q=(\mathrm{I}-A)^{-1}(\mathrm{I}+A)$ is an orthogonal matrix. Can you prove that $\mathrm{I}-A$ is always invertible?
Redo Example 4.43 with the dot product replaced by the weighted inner product$$\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+2 v_2 w_2+3 v_3 w_3+4 v_4 w_4 \text {. }$$
Find the monic Laguerre polynomials of degrees 4 and 5 and their norms.
Let $V=\mathcal{P}^{(1)}$ be the vector space consisting of linear polynomials $p(t)=a t+b$.(a) Carefully explain why $\langle p, q\rangle=\int_0^1 t p(t) q(t) d t$ defines an inner product on $V$.(b) Find all polynomials $p(t)=a t+b \in V$ that are orthogonal to $p_1(t)=1$ based on this inner product. (c) Use part (b) to construct an orthonormal basis of $V$ for this inner product. (d) Find an orthonormal basis of the space $\mathcal{P}^{(2)}$ of quadratic polynomials for the same inner product. Hint: First find a quadratic polynomial that is orthogonal to the basis you constructed in part $(c)$.
Repeat Exercise 4.2 .17 using the numerically stable algorithm (4.28) and check that you get the same result. Which of the two algorithms was easier for you to implement?
Suppose $S$ is an $n \times n$ matrix whose columns form an orthogonal, but not orthonormal, basis of $\mathbb{R}^n$. (a) Find a formula for $S^{-1}$ mimicking the formula $Q^{-1}=Q^T$ for an orthogonal matrix. (b) Use your formula to determine the inverse of the wavelet matrix $W$ whose columns form the orthogonal wavelet basis (4.9) of $\mathbb{R}^4$.
Prove that the orthogonal complement $W^{\perp}$ of a subspace $W \subset V$ is itself a subspace.$\dagger$ In general, a subset $W \subset V$ of a normed vector space is dense if, for every $\mathbf{v} \in V$, and every $\varepsilon>0$, one can find $\mathbf{w} \in W$ with $\|\mathbf{v}-\mathbf{w}\|<\varepsilon$. The Weierstrass Approximation Theorem, [19; Theorem 10.2.2], tells us that the polynomials form a dense subspace of the space of continuous functions, and underlies the proof of the result mentioned in the preceding paragraph.
Prove the integration formula (4.67).
Explain why the functions $\cos x, \sin x$ form an orthogonal basis for the space of solutions to the differential equation $y^{\prime \prime}+y=0$ under the $\mathrm{L}^2$ inner product on $[-\pi, \pi]$.
Redo each of the exercises in the preceding subsection by implementing the numerically stable Gram-Schmidt process (4.28) instead, and verify that you end up with the same orthonormal basis.
Let $\mathbf{v}_1, \ldots, \mathbf{v}_n$ and $\mathbf{w}_1, \ldots, \mathbf{w}_n$ be two sets of linearly independent vectors in $\mathbb{R}^n$. Show that all their dot products are the same, so $\mathbf{v}_i \cdot \mathbf{v}_j=\mathbf{w}_i \cdot \mathbf{w}_j$ for all $i, j=1, \ldots, n$, if and only if there is an orthogonal matrix $Q$ such that $\mathbf{w}_i=Q \mathbf{v}_i$ for all $i=1, \ldots, n$.
Let $V=\mathcal{P}^{(4)}$ denote the space of quartic polynomials, with the $\mathrm{L}^2$ inner product $\langle p, q\rangle=\int_{-1}^1 p(x) q(x) d x$. Let $W=\mathcal{P}^{(2)}$ be the subspace of quadratic polynomials.(a) Write down the conditions that a polynomial $p \in \mathcal{P}^{(4)}$ must satisfy in order to belong to the orthogonal complement $W^{\perp}$. (b) Find a basis for and the dimension of $W^{\perp}$.(c) Find an orthogonal basis for $W^{\perp}$.
(a) The physicists' Hermite polynomials are orthogonal with respect to the inner product $\langle f, g\rangle=\int_{-\infty}^{\infty} f(t) g(t) e^{-t^2} d t$. Find the first five monic Hermite polynomials. (b) The probabilists prefer to use the inner product $\langle f, g\rangle=\int_{-\infty}^{\infty} f(t) g(t) e^{-t^2 / 2} d t$. Find the first five of their monic Hermite polynomials.(c) Can you find a change of variables that transforms the physicists' versions to the probabilists' versions?
Do the functions $e^{x / 2}, e^{-x / 2}$ form an orthogonal basis for the space of solutions to the differential equation $4 y^{\prime \prime}-y=0$ under the $\mathrm{L}^2$ inner product on $[0,1]$ ? If not, can you find an orthogonal basis of the solution space?
Prove that (4.28) does indeed produce an orthonormal basis. Explain why the result is the same orthonormal basis as the ordinary Gram-Schmidt method.
Suppose $\mathbf{u}_1, \ldots, \mathbf{u}_k$ form an orthonormal set of vectors in $\mathbb{R}^n$ with $k<n$. Let $Q=\left(\mathbf{u}_1 \mathbf{u}_2 \ldots \mathbf{u}_k\right)$ denote the $n \times k$ matrix whose columns are the orthonormal vectors.(a) Prove that $Q^T Q=\mathrm{I}_k$. (b) Is $Q Q^T=\mathrm{I}_n$ ?
Let $W \subset V$. Prove that (a) $W \cap W^{\perp}=\{0\}, \quad(b) W \subseteq\left(W^{\perp}\right)^{\perp}$.
The Chebyshev polynomials: (a) Prove that $T_n(t)=\cos (n \arccos t), n=0,1,2, \ldots$, form a system of orthogonal polynomials under the weighted inner product$$\langle f, g\rangle=\int_{-1}^1 \frac{f(t) g(t) d t}{\sqrt{1-t^2}} .$$(b) What is $\left\|T_n\right\|$ ? (c) Write out the formulas for $T_0(t), \ldots, T_6(t)$ and plot their graphs.
(a) Prove that the vectors $\mathbf{v}_1=(1,1,1)^T, \mathbf{v}_2=(1,1,-2)^T, \mathbf{v}_3=(-1,1,0)^T$, form an orthogonal basis of $\mathbb{R}^3$ with the dot product. (b) Use orthogonality to write the vector $\mathbf{v}=(1,2,3)^T$ as a linear combination of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$. (c) Verify the formula (4.8) for $\|\mathbf{v}\|$. (d) Construct an orthonormal basis, using the given vectors. (e) Write $\mathbf{v}$ as a linear combination of the orthonormal basis, and verify (4.5).
Let $\mathbf{w}_j^{(j)}$ be the vectors in the stable Gram-Schmidt algorithm (4.28). Prove that the coefficients in (4.23) are given by $r_{i i}=\left\|\mathbf{w}_i^{(i)}\right\|$, and $r_{i j}=\left\langle\mathbf{w}_j^{(i)}, \mathbf{u}_i\right\rangle$ for $i<j$.
Let $\mathbf{u}_1, \ldots, \mathbf{u}_n$ and $\hat{\mathbf{u}}_1, \ldots, \hat{\mathbf{u}}_n$ be orthonormal bases of an inner product space $V$. Prove that $\widehat{\mathbf{u}}_i=\sum_{j=1}^n q_{i j} \mathbf{u}_j$ for $i=1, \ldots, n$, where $Q=\left(q_{i j}\right)$ is an orthogonal matrix.
Let $V$ be an inner product space. Prove that (a) $V^{\perp}=\{0\}$, (b) $\{0\}^{\perp}=V$.
Does the Gram-Schmidt process for the inner product (4.75) lead to the Chebyshev polynomials $T_n(t)$ defined in the preceding exercise? Explain why or why not.
(a) Prove that $\mathbf{v}_1=\left(\frac{3}{5}, 0, \frac{4}{5}\right)^T, \mathbf{v}_2=\left(-\frac{4}{13}, \frac{12}{13}, \frac{3}{13}\right)^T, \mathbf{v}_3=\left(-\frac{48}{65},-\frac{5}{13}, \frac{36}{65}\right)^T$, form an orthonormal basis for $\mathbb{R}^3$ for the usual dot product. (b) Find the coordinates of $\mathbf{v}=(1,1,1)^T$ relative to this basis. (c) Verify formula (4.5) in this particular case.
Let $A$ be an $m \times n$ matrix whose columns are nonzero, mutually orthogonal vectors in $\mathbb{R}^m$. (a) Explain why $m \geq n$. (b) Prove that $A^T A$ is a diagonal matrix. What are the diagonal entries? (c) Is $\bar{A} A^T$ diagonal?
Prove that if $W_1 \subset W_2$ are finite-dimensional subspaces of an inner product space, then $W_1^{\perp} \supset W_2^{\perp}$.
Find two functions that form an orthogonal basis for the space of the solutions to the differential equation $y^{\prime \prime}-3 y^{\prime}+2 y=0$ under the $\mathrm{L}^2$ inner product on $[0,1]$.
Let $\mathbb{R}^2$ have the inner product defined by the positive definite matrix $K=\left(\begin{array}{rr}2 & -1 \\ -1 & 3\end{array}\right)$. (a) Show that $\mathbf{v}_1=(1,1)^T, \mathbf{v}_2=(-2,1)^T$ form an orthogonal basis. (b) Write the vector $\mathbf{v}=(3,2)^T$ as a linear combination of $\mathbf{v}_1, \mathbf{v}_2$ using the orthogonality formula (4.7). (c) Verify the formula (4.8) for $\|\mathbf{v}\|$. (d) Find an orthonormal basis $\mathbf{u}_1, \mathbf{u}_2$ for this inner product. (e) Write $\mathbf{v}$ as a linear combination of the orthonormal basis, and verify (4.5).
Let $K>0$ be a positive definite $n \times n$ matrix. Prove that an $n \times n$ matrix $S$ satisfies $S^T K S=\mathrm{I}$ if and only if the columns of $S$ form an orthonormal basis of $\mathbb{R}^n$ with respect to the inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T K \mathbf{w}$.
(a) Show that if $W, Z \subset \mathbb{R}^n$ are complementary subspaces, then $W^{\perp}$ and $Z^{\perp}$ are also complementary subspaces. (b) Sketch a picture illustrating this result when $W$ and $Z$ are lines in $\mathbb{R}^2$.
Find an orthogonal basis for the space of solutions to the differential equation $y^{\prime \prime \prime}-y^{\prime \prime}+y^{\prime}-y=0$ for the $\mathrm{L}^2$ inner product on $[-\pi, \pi]$.
(a) Let $\mathbf{u}_1, \ldots, \mathbf{u}_n$ be an orthonormal basis of a finite-dimensional inner product space $V$. Let $\mathbf{v}=c_1 \mathbf{u}_1+\cdots+c_n \mathbf{u}_n$ and $\mathbf{w}=d_1 \mathbf{u}_1+\cdots+d_n \mathbf{u}_n$ be any two elements of $V$. Prove that $\langle\mathbf{v}, \mathbf{w}\rangle=c_1 d_1+\cdots+c_n d_n$.(b) Write down the corresponding inner product formula for an orthogonal basis.
Groups: A set of $n \times n$ matrices $G \subset \mathcal{M}_{n \times n}$ is said to form a group if(1) whenever $A, B \in G$, so is the product $A B \in G$, and(2) whenever $A \in G$, then $A$ is nonsingular, and $A^{-1} \in G$.
Prove that if $W, Z$ are subspaces of an inner product space, then $(W+Z)^{\perp}=W^{\perp} \cap Z^{\perp}$. (See Exercise 2.2.22(b) for the definition of the sum of two subspaces.)
In this exercise, we investigate the effect of more general changes of variables on orthogonal polynomials. (a) Prove that $t=2 s^2-1$ defines a one-to-one map from the interval $0 \leq s \leq 1$ to the interval $-1 \leq t \leq 1$. (b) Let $p_k(t)$ denote the monic Legendre polynomials, which are orthogonal on $-1 \leq t \leq 1$. Show that $q_k(s)=p_k\left(2 s^2-1\right)$ defines a polynomial. Write out the cases $k=0,1,2,3$ explicitly. (c) Are the polynomials $q_k(s)$ orthogonal under the $\mathrm{L}^2$ inner product on $[0,1]$ ? If not, do they retain any sort of orthogonality property?
Find an example that demonstrates why equation (4.5) is not valid for a nonorthonormal basis.
Unitary matrices: A complex, square matrix $U$ is called unitary if it satisfies $U^{\dagger} U=\mathrm{I}$, where $U^{\dagger}=\overline{U^T}$ denotes the Hermitian adjoint in which one first transposes and then takes complex conjugates of all entries. (a) Show that $U$ is a unitary matrix if and only if $U^{-1}=U^{\dagger}$. (b) Show that the following matrices are unitary and compute their inverses:(i) $\left(\begin{array}{ll}\frac{1}{\sqrt{2}} & \frac{\mathrm{i}}{\sqrt{2}} \\ \frac{\mathrm{i}}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{array}\right)$,(ii) $\left(\begin{array}{ccc}\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} & -\frac{1}{2 \sqrt{3}}+\frac{1}{2} & -\frac{1}{2 \sqrt{3}}-\frac{1}{2} \\ \frac{1}{\sqrt{3}} & -\frac{1}{2 \sqrt{3}}-\frac{\mathrm{i}}{2} & -\frac{1}{2 \sqrt{3}}+\frac{\mathrm{i}}{2}\end{array}\right)$,(iii)$$\left(\begin{array}{rrrr}\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\\frac{1}{2} & \frac{i}{2} & -\frac{1}{2} & -\frac{i}{2} \\\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \\\frac{1}{2} & -\frac{i}{2} & -\frac{1}{2} & \frac{i}{2}\end{array}\right)$$(c) Are the following matrices unitary?(i) $\left(\begin{array}{cc}2 & 1+2 \mathrm{i} \\ 1-2 \mathrm{i} & 3\end{array}\right)$,(ii) $\frac{1}{5}\left(\begin{array}{cc}-1+2 \mathrm{i} & -4-2 \mathrm{i} \\ 2-4 \mathrm{i} & -2-\mathrm{i}\end{array}\right)$,(iii) $\left(\begin{array}{rr}\frac{12}{13} & \frac{5}{13} \\ \frac{5}{13} & -\frac{12}{13}\end{array}\right)$.(d) Show that $U$ is a unitary matrix if and only if its columns form an orthonormal basis of $\mathbb{C}^n$ with respect to the Hermitian dot product. (e) Prove that the set of unitary matrices forms a group, as defined in Exercise 4.3.24.
Fill in the details of the proof of Proposition 4.44.
(a) Show that the change of variables $s=e^{-t}$ maps the Laguerre inner product (4.66) to the standard $\mathrm{L}^2$ inner product on $[0,1]$. However, explain why this does not allow you to change Legendre polynomials into Laguerre polynomials. (b) Describe the functions resulting from applying the change of variables to the modified Legendre polynomials (4.74) and their orthogonality properties. (c) Describe the functions that result from applying the inverse change of variables to the Laguerre polynomials (4.68) and their orthogonality properties.
Use orthogonality to write the polynomials $1, x$ and $x^2$ as linear combinations of the orthogonal basis (4.1).
Write down the $Q R$ matrix factorization corresponding to the vectors in Example 4.17.
Prove Lemma 4.35 .
Explain how to adapt the numerically stable Gram-Schmidt method in (4.28) to construct a system of orthogonal polynomials. Test your algorithm on one of the preceding exercises.
(a) Prove that the polynomials $P_0(t)=1, P_1(t)=t, P_2(t)=t^2-\frac{1}{3}, P_3(t)=t^3-\frac{3}{5} t$, form an orthogonal basis for the vector space $\mathcal{P}^{(3)}$ of cubic polynomials for the $\mathrm{L}^2$ inner product $\langle f, g\rangle=\int_{-1}^1 f(t) g(t) d t$. (b) Find an orthonormal basis of $\mathcal{P}^{(3)}$. (c) Write $t^3$ as a linear combination of $P_0, P_1, P_2, P_3$ using the orthogonal basis formula (4.7).
Find the $Q R$ factorization of the following matrices: (a) $\left(\begin{array}{rr}1 & -3 \\ 2 & 1\end{array}\right)$,(b) $\left(\begin{array}{ll}4 & 3 \\ 3 & 2\end{array}\right)$,(c) $\left(\begin{array}{rrr}2 & 1 & -1 \\ 0 & 1 & 3 \\ -1 & -1 & 1\end{array}\right)$,(d) $\left(\begin{array}{rrr}0 & 1 & 2 \\ -1 & 1 & 1 \\ -1 & 1 & 3\end{array}\right)$,(e) $\left(\begin{array}{rrr}0 & 0 & 2 \\ 0 & 4 & 1 \\ -1 & 0 & 1\end{array}\right)$,(f) $\left(\begin{array}{llll}1 & 1 & 1 & 1 \\ 1 & 2 & 1 & 0 \\ 1 & 1 & 2 & 1 \\ 1 & 0 & 1 & 1\end{array}\right)$.
Let $W \subset V$ with $\operatorname{dim} V=n$. Suppose $\mathbf{w}_1, \ldots, \mathbf{w}_m$ is an orthogonal basis for $W$ and $\mathbf{w}_{m+1}, \ldots, \mathbf{w}_n$ is an orthogonal basis for $W^{\perp}$. (a) Prove that the combination $\mathbf{w}_1, \ldots, \mathbf{w}_n$ forms an orthogonal basis of $V$. (b) Show that if $\mathbf{v}=c_1 \mathbf{w}_1+\cdots+c_n \mathbf{w}_n$ is any vector in $V$, then its orthogonal decomposition $\mathbf{v}=\mathbf{w}+\mathbf{z}$ is given by $\mathbf{w}=c_1 \mathbf{w}_1+\cdots+c_m \mathbf{w}_m \in W$ and $\mathbf{z}=c_{m+1} \mathbf{w}_{m+1}+\cdots+c_n \mathbf{w}_n \in W^{\perp}$.
(a) Prove that the polynomials $P_0(t)=1, P_1(t)=t-\frac{2}{3}, P_2(t)=t^2-\frac{6}{5} t+\frac{3}{10}$, form an orthogonal basis for $\mathcal{P}^{(2)}$ with respect to the weighted inner product $\langle f, g\rangle=\int_0^1 f(t) g(t) t d t$. (b) Find the corresponding orthonormal basis.(c) Write $t^2$ as a linear combination of $P_0, P_1, P_2$ using the orthogonal basis formula (4.7).
For each of the following linear systems, find the $Q R$ factorization of the coefficient matrix, and then use your factorization to solve the system: $(i)\left(\begin{array}{rr}1 & 2 \\ -1 & 3\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{r}-1 \\ 2\end{array}\right)$,(ii) $\left(\begin{array}{rrr}2 & 1 & -1 \\ 1 & 0 & 2 \\ 2 & -1 & 3\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}2 \\ -1 \\ 0\end{array}\right)$,(iii) $\left(\begin{array}{rrr}1 & 1 & 0 \\ -1 & 0 & 1 \\ 0 & -1 & 1\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$.
Consider the subspace $W=\{u(a)=0=u(b)\}$ of the vector space $\mathrm{C}^0[a, b]$ with the usual $\mathrm{L}^2$ inner product. (a) Show that $W$ has a complementary subspace of dimension 2 . (b) Prove that there does not exist an orthogonal complement of $W$. Thus, an infinitedimensional subspace may not admit an orthogonal complement!
Write the following trigonometric polynomials in terms of the basis functions (4.11):(a) $\cos ^2 x$,(b) $\cos x \sin x$(c) $\sin ^3 x$,(d) $\cos ^2 x \sin ^3 x$(e) $\cos ^4 x$.
Use the numerically stable version of the Gram-Schmidt process to find the $Q R$ factorizations of the $3 \times 3,4 \times 4$ and $5 \times 5$ versions of the tridiagonal matrix that has 4 's along the diagonal and 1's on the sub- and super-diagonals, as in Example 1.37.
For each of the following matrices $A,(i)$ find a basis for each of the four fundamental subspaces; (ii) verify that the image and cokernel are orthogonal complements; (iii) verify that the coimage and kernel are orthogonal complements:(a) $\left(\begin{array}{ll}1 & -2 \\ 2 & -4\end{array}\right)$,(b) $\left(\begin{array}{ll}5 & 0 \\ 1 & 2 \\ 0 & 2\end{array}\right)$,(c) $\left(\begin{array}{rrr}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right)$,(d) $\left(\begin{array}{rrrr}1 & 2 & 0 & 1 \\ -1 & 1 & 3 & 1 \\ 0 & 3 & 3 & 2\end{array}\right)$,(e) $\left(\begin{array}{rrrrr}3 & 1 & 4 & 2 & 7 \\ 1 & 1 & 2 & 0 & 3 \\ 5 & 2 & 7 & 3 & 12\end{array}\right)$,(f) $\left(\begin{array}{rrrr}1 & 3 & 0 & -2 \\ -2 & 1 & 2 & 3 \\ -3 & 5 & 4 & 4 \\ 1 & -4 & -2 & -1\end{array}\right)$,(g) $\left(\begin{array}{rrrr}-1 & 2 & 2 & -1 \\ 2 & -4 & -5 & 2 \\ -3 & 6 & 2 & -3 \\ 1 & -2 & -3 & 1 \\ -2 & 4 & -5 & -2\end{array}\right)$.
Write down an orthonormal basis of the space of trigonometric polynomials $\mathcal{T}^{(n)}$ with respect to the $\mathrm{L}^2$ inner product $\langle f, g\rangle=\int_{-\pi}^\pi f(x) g(x) d x$.
Prove that the $Q R$ factorization of a matrix is unique if all the diagonal entries of $R$ are assumed to be positive.
For each of the following matrices, use Gaussian elimination on the augmented matrix $(A \mid \mathbf{b})$ to determine a basis for its cokernel:(a) $\left(\begin{array}{ll}9 & -6 \\ 6 & -4\end{array}\right)$,(b) $\left(\begin{array}{rr}1 & 3 \\ 2 & 6 \\ -3 & -9\end{array}\right)$,(c) $\left(\begin{array}{rrr}1 & 1 & 3 \\ -1 & 1 & -2 \\ -1 & 3 & 6\end{array}\right)$,(d) $\left(\begin{array}{rrr}1 & -2 & -2 \\ 0 & -1 & 3 \\ 2 & -5 & -1 \\ -2 & 2 & 10\end{array}\right)$.
Show that the $2 n+1$ complex exponentials $e^{i k x}$ for $k=-n,-n+1, \ldots,-1,0,1, \ldots, n$, form an orthonormal basis for the space of complex-valued trigonometric polynomials under the Hermitian inner product $\langle f, g\rangle=\frac{1}{2 \pi} \int_{-\pi}^\pi f(x) \overline{g(x)} d x$.
(a) How many arithmetic operations are required to compute the $Q R$ factorization of an $n \times n$ matrix? (b) How many additional operations are needed to utilize the factorization to solve a linear system $A \mathbf{x}=\mathbf{b}$ via (4.34)? (c) Compare the amount of computational effort with standard Gaussian Elimination.
Let $A=\left(\begin{array}{rrrr}1 & -2 & 2 & -1 \\ -2 & 4 & -3 & 5 \\ -1 & 2 & 0 & 7\end{array}\right)$.(a) Find a basis for coimg $A$. (b) Use Theorem 4.49 to find a basis of $\operatorname{img} A$. (c) Write each column of $A$ as a linear combination of the basis vectors you found in part $(b)$.$\uparrow$ An alternative is to orthogonally project the general solution onto the coimage. The result is the same.
Prove the trigonometric integral identities (4.13).
Suppose $A$ is an $m \times n$ matrix with $\operatorname{rank} A=n$. (a) Show that applying the GramSchmidt algorithm to the columns of $A$ produces an orthonormal basis for img $A$. (b) Prove that this is equivalent to the matrix factorization $A=Q R$, where $Q$ is an $m \times n$ matrix with orthonormal columns, while $R$ is a nonsingular $n \times n$ upper triangular matrix. (c) Show that the $Q R$ program in the text also works for rectangular, $m \times n$, matrices as stated, the only modification being that the row indices $i$ run from 1 to $m$. (d) Apply this method to factor(i) $\left(\begin{array}{rr}1 & -1 \\ 2 & 3 \\ 0 & 2\end{array}\right)$,(ii) $\left(\begin{array}{rr}-3 & 2 \\ 1 & -1 \\ 4 & 1\end{array}\right)$,(iii)$\left(\begin{array}{rr}-1 & 1 \\ 1 & -2 \\ -1 & -3 \\ 0 & 5\end{array}\right)$(iv)$$\left(\begin{array}{rrr}0 & 1 & 2 \\-3 & 1 & -1 \\-1 & 0 & -2 \\1 & 1 & -2\end{array}\right)$$(e) Explain what happens if $\operatorname{rank} A<n$.
Write down the compatibility conditions on the following systems of linear equations by first computing a basis for the cokernel of the coefficient matrix. (a) $2 x+y=a$, $x+4 y=b,-3 x+2 y=c$;(b) $x+2 y+3 z=a,-x+5 y-2 z=b, 2 x-3 y+5 z=c$;(c) $x_1+2 x_2+3 x_3=b_1, x_2+2 x_3=b_2, 3 x_1+5 x_2+7 x_3=b_3,-2 x_1+x_2+4 x_3=b_4$;(d) $x-3 y+2 z+w=a, 4 x-2 y+2 z+3 w=b, 5 x-5 y+4 z+4 w=c, 2 x+4 y-2 z+w=d$.
Fill in the complete details of the proof of Theorem 4.9.
(a) According to Exercise 4.2.14, the Gram-Schmidt process can also be applied to produce orthonormal bases of complex vector spaces. In the case of $\mathbb{C}^n$, explain how this is equivalent to the factorization of a nonsingular complex matrix $A=U R$ into the product of a unitary matrix $U$ (see Exercise 4.3.25) and a nonsingular upper triangular matrix $R$.(b) Factor the following complex matrices into unitary times upper triangular:(i) $\left(\begin{array}{rr}\mathrm{i} & 1 \\ -1 & 2 \mathrm{i}\end{array}\right)$,(ii)$$\left(\begin{array}{cc}1+i & 2-i \\1-i & -i\end{array}\right)$$(iii) $\left(\begin{array}{lll}\mathrm{i} & 1 & 0 \\ 1 & \mathrm{i} & 1 \\ 0 & 1 & \mathrm{i}\end{array}\right)$,(iv)$$\left(\begin{array}{ccc}\mathrm{i} & 1 & -\mathrm{i} \\1-\mathrm{i} & 0 & 1+\mathrm{i} \\-1 & 2+3 \mathrm{i} & 1\end{array}\right)$$(c) What can you say about uniqueness of the factorization?
For each of the following $m \times n$ matrices, decompose the first standard basis vector $\mathbf{e}_1=\mathbf{w}+\mathbf{z} \in \mathbb{R}^n$, where $\mathbf{w} \in \operatorname{coimg} A$ and $\mathbf{z} \in \operatorname{ker} A$. Verify your answer by expressing $\mathbf{w}$ as a linear combination of the rows of $A$.(a) $\left(\begin{array}{lll}1 & -2 & 1 \\ 2 & -3 & 2\end{array}\right)$,(b) $\left(\begin{array}{rrr}1 & 1 & 2 \\ -1 & 0 & -1 \\ -2 & -1 & -3\end{array}\right)$,(c) $\left(\begin{array}{rrrr}1 & -1 & 0 & 3 \\ 2 & 1 & 3 & 3 \\ 1 & 2 & 3 & 0\end{array}\right)$,(d) $\left(\begin{array}{rrrrr}-1 & 1 & 1 & -1 & 2 \\ -3 & 2 & -1 & -2 & 0\end{array}\right)$.
(a) Write down the Householder matrices corresponding to the following unit vectors:(i) $(1,0)^T$,(ii) $\left(\frac{3}{5}, \frac{4}{5}\right)^T$,(iii)$(0,1,0)^T,(i v)\left(\frac{1}{\sqrt{2}}, 0,-\frac{1}{\sqrt{2}}\right)^T$.(b) Find all vectors fixed by a Householder matrix, i.e., $H \mathbf{v}=\mathbf{v}$ - first for the matrices in part (a), and then in general. (c) Is a Householder matrix a proper or improper orthogonal matrix?
For each of the following linear systems, (i) verify compatibility using the Fredholm alternative, (ii) find the general solution, and (iii) find the solution of minimum Euclidean norm:$$\text { (a) } \begin{aligned}2 x-4 y & =-6, \\-x+2 y & =3,\end{aligned}$$$$x+3 y+5 z=3 \text {, }$$(d)$$\begin{aligned}& -x+4 y+9 z=11, \\& 2 x+3 y+4 z=0,\end{aligned}$$(e)(b)$$2 x+3 y=-1,$$(c)$$\begin{aligned}6 x-3 y+9 z & =12, \\2 x-y+3 z & =4, \\& x-y+2 z+3 w=5\end{aligned}$$$$2 x-y+3 z=4,$$$$\begin{aligned}x_1-3 x_2+7 x_3 & =-8, \\2 x_1+x_2 & =5,\end{aligned}$$$$x-y+2 z+3 w=5,$$(f)$$\begin{aligned}3 x-3 y+5 z+7 w & =13, \\-2 x+2 y+z+4 w & =0 .\end{aligned}$$
Use Householder's Method to solve Exercises 4.3.27 and 4.3.29.
Show that if $A=A^T$ is a symmetric matrix, then $A \mathbf{x}=\mathbf{b}$ has a solution if and only if $\mathbf{b}$ is orthogonal to $\operatorname{ker} A$.
Let $H_n=Q_n R_n$ be the $Q R$ factorization of the $n \times n$ Hilbert matrix (1.72). (a) Find $Q_n$ and $R_n$ for $n=2,3,4$. (b) Use a computer to find $Q_n$ and $R_n$ for $n=10$ and 20 . (c) Let $\mathbf{x}^{\star} \in \mathbb{R}^n$ denote the vector whose $i^{\text {th }}$ entry is $x_i^{\star}=(-1)^i i /(i+1)$. For the values of $n$ in parts (a) and (b), compute $\mathbf{y}^{\star}=H_n \mathbf{x}^{\star}$. Then solve the system $H_n \mathbf{x}=\mathbf{y}^{\star}(i)$ directly using Gaussian Elimination; (ii) using the $Q R$ factorization based on(4.34); (iii) using Householder's Method. Compare the results to the correct solution $\mathbf{x}^{\star}$ and discuss the pros and cons of each method.
Suppose $\mathbf{v}_1, \ldots, \mathbf{v}_n$ span a subspace $V \subset \mathbb{R}^m$. Prove that $\mathbf{w}$ is orthogonal to $V$ if and only if $\mathbf{w} \in \operatorname{coker} A$, where $A=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$ is the matrix with the indicated columns.
Write out a pseudocode program to implement Householder's Method. The input should be an $n \times n$ matrix $A$ and the output should be the Householder unit vectors $\mathbf{u}_1, \ldots, \mathbf{u}_{n-1}$ and the upper triangular matrix $R$. Test your code on one of the examples in Exercises 4.3.26-28.
Let $A=\left(\begin{array}{rrrr}1 & -1 & 0 & 2 \\ 2 & -2 & 0 & 4 \\ -1 & 1 & 1 & -1 \\ 0 & 0 & 2 & 2\end{array}\right)$.(a) Find an orthogonal basis for coimg $A$.(b) Find anorthogonal basis for ker $A$. (c) If you combine your bases from parts (a) and (b), do you get an orthogonal basis of $\mathbb{R}^4$ ? Why or why not?
Prove that if $\mathbf{v}_1, \ldots, \mathbf{v}_r$ are a basis of coimg $A$, then their images $A \mathbf{v}_1, \ldots, A \mathbf{v}_r$ are a basis for img $A$.
True or false: The standard algorithm for finding a basis for ker $A$ will always produce an orthogonal basis.
Is Theorem 4.45 true as stated for complex matrices? If not, can you formulate a similar theorem that is true? What is the Fredholm alternative for complex matrices?