$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 14.7: Review Problems

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

1. Let $$D = \begin{pmatrix}\lambda_{1} & 0 \\ 0 & \lambda_{2}\end{pmatrix}$$

a) Write $$D$$ in terms of the vectors $$e_{1}$$ and $$e_{2}$$, and their transposes.

b) Suppose $$P = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ is invertible. Show that $$D$$ is similar to

$$M = \frac{1}{ad - bc}\begin{pmatrix}\lambda_{1}ad - \lambda_{2}bc & -(\lambda_{1} - \lambda_{2})ab \\ (\lambda_{1} - \lambda_{2})cd & -\lambda{1}bc + \lambda_{2}ad\end{pmatrix}$$

c) Suppose the vectors $$(a,b)$$ and $$(c,d)$$ are orthogonal. What can you say about $$M$$ in this case? ($$\textit{Hint:}$$ think about what $$M^{T}$$ is equal to.)

2. Suppose $$S = {v_{1},...,v_{n}}$$ is an $$\textit{orthogonal}$$ (not orthonormal) basis for $$\mathbb{R}^{n}$$. Then we can write any vector $$v$$ as $$v = \sum_{i} c^{i}v_{i}$$ for some constants $$c^{i}$$. Find a formula for the constants $$c^{i}$$ in terms of $$v$$ and the vectors in $$S$$.

3. Let $$u, v$$ be linearly independent vectors in $$\mathbb{R}^{3}$$, and $$P = span{u, v}$$ be the plane spanned by $$u$$ and $$v$$.

(a) Is the vector $$v^{\perp} := v - \frac{u \cdot v}{u \cdot u}u$$ in the plane $$P$$?

(b) What is the (cosine of the) angle between $$v^{\perp}$$ and $$u^{\perp}$$?

(c) How can you find a third vector perpendicular to both $$u$$ and $$v^{\perp}$$?

(d) Construct an orthonormal basis for $$\mathbb{R}^{3}$$ from $$u$$ and $$v$$.

(e) Test your abstract formulæ starting with $$u = (1,2,0)$$ and $$v = (0,1,1)$$.

4. Find an orthonormal basis for $$\mathbb{R}^{4}$$ which includes $$(1,1,1,1)$$ using the following procedure:

(a) Pick a vector perpendicular to the vector

$$v_{1} = \begin{pmatrix}1\\1\\1\\1\end{pmatrix}$$

from the solution set of the matrix equation

$$v_{1}^{T}x = 0.$$

Pick the vector $$v_{2}$$ obtained from the standard Gaussian elimination procedure which is the coefficient of $$x_{2}$$.

(b) Pick a vector perpendicular to both $$v_{1}$$ and $$v_{2}$$ from the solutions set of the matrix equation

$$\begin{pmatrix}v_{1}^{T} \\ v_{2}^{T}\end{pmatrix}x = 0.$$

Pick the vector $$v_{3}$$ obtained from the standard Gaussian elimination procedure with $$x_{3}$$ as the coefficient.

(c) Pick a vector perpendicular to $$v_{1}, v_{2}$$, and $$v_{3}$$ from the solution set of the matrix equation

$$\begin{pmatrix}v_{1}^{T} \\ v_{2}^{T} \\ v_{3}^{T}\end{pmatrix}x = 0. Pick the vector $$v_{4}$$ obtained from the standard Gaussian elimination procedure with $$x_{3}$$ as the coefficient. (d) Normalize the four vectors obtained above. 5. Use the inner product$$f \cdot g := \int_{0}^{1} f(x)g(x)dx$$on the vector space $$V =span{1,x,x^{2},x^{3}}$$ to perform the Gram-Schmidt procedure on the set of vectors $${1,x,x^{2},x^{3}}$$ 6. Use the inner product on the vector space $$V = span{sin(x), sin(2x), sin(3x)}$$ to perform the Gram-Schmidt procedure on the set of vectors $${sin(x), sin(2x), sin(3x)}$$. What do you suspect about the vector space $$span{sin(nx) | n \in N}$$? What do you suspect about the vector space $$span{sin(ax) | a \in R}$$? 7. 1. Show that if $$Q$$ is an orthogonal $$n \times n$$ matrix then$$u \cdot v = (Qu) \cdot (Qv),$$for any $$u, v \in \mathbb{R}^{n}$$. That is, $$Q$$ preserves the inner product. 2. Does $$Q$$ preserve the outer product? 3. If $${u_{1},...,u_{n}}$$ is an orthonormal set and $${\lambda_{1},··· , \lambda_{n}}$$ is a set of numbers then what are the eigenvalues and eigenvectors of the matrix $$M = \sum^{n}_{i=1} \lambda_{i}u_{i}u^{T}_{i}$$? 4. How does $$Q$$ change this matrix? How do the eigenvectors and eigenvalues change? 8. Carefully write out the Gram-Schmidt procedure for the set of vectors$$\begin{Bmatrix}\begin{pmatrix}1 \\1 \\1\end{pmatrix}, \begin{pmatrix}1 \\-1 \\1\end{pmatrix}, \begin{pmatrix}1 \\1 \\-1\end{pmatrix}\end{Bmatrix}.$$Are you free to rescale the second vector obtained in the procedure to a vector with integer components? 9. a) Suppose $$u$$ and $$v$$ are linearly independent. Show that $$u$$ and $$v^{\perp}$$ are also linearly independent. Explain why $${u, v^{\perp}}$$ is a basis for $$span{u,v}$$. b) Repeat the previous problem, but with three independent vectors $$u, v, w$$. 10. Find the $$QR$$ factorization of$$M = \begin{pmatrix}1&0&2\\-1&2&0\\-1&-2&2\end{pmatrix}.$$11. Given any three vectors $$u, v, w$$, when do $$v^{\perp}$$ or $$w^{\perp}$$ of the Gram-Schmidt procedure vanish? 12. For $$U$$ a subspace of $$W$$, use the subspace theorem to check that $$U^{\perp}$$ is a subspace of $$W$$. 13. Let $$S_{n}$$ and $$A_{n}$$ define the space of $$n \times n$$ symmetric and anti-symmetric matrices respectively. These are subspaces of the vector space $$M^{n}_{n}$$ of all $$n \times n$$ matrices. What is $$dim M_{n}^{n}$$, $$dim S_{n}$$ and $$dim A_{n}$$? Show that $$M^{n}_{n} = S_{n} + A_{n}$$. Is $$A^{\perp}_{n} = S_{n}$$? Is $$M^{n}_{n} = S_{n} \oplus A_{n}$$? 14. The vector space $$V = span{sin(t), sin(2t), sin(3t)}$$ has an inner product:$$f \cdot g := \int_{0}^{2\pi} f(t)g(t)dt. Find the orthogonal compliment to $$U = span{sin(t) + sin(2t)}$$ in $$V$$. Express $$sin(t) - sin(2t)$$ as the sum of vectors from $$U$$ and $$U^{T}$$.