14.7: Review Problems

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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 $u^{\perp}$?

(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.

\[\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 $span{sin(ax) | a \in R}$?

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.
Does $Q$ preserve the outer product?
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 $M = \sum^{n}_{i=1} \lambda_{i}u_{i}u^{T}_{i}$?
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?

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 $U^{T}$.

Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)

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