$\newcommand{\R}{\mathbb R }$ $\newcommand{\bfa}{\mathbf a}$ $\newcommand{\bfb}{\mathbf b}$ $\newcommand{\bff}{\mathbf f}$ $\newcommand{\bfu}{\mathbf u}$ $\newcommand{\bfx}{\mathbf x}$ $\newcommand{\bfy}{\mathbf y}$ $\newcommand{\ep}{\varepsilon}$

Problems for Chapter 0: Preliminaries

Basic Skills

1. You should have mastered standard pre-calculus material such as sketching ellipses and hyperbolas: $$ \frac {(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} = r^2 $$ $$ \frac {(x - x_0)^2}{a^2} - \frac{(y-y_0)^2}{b^2} = r^2. $$

2. You should be able to answer simple questions that are based on knowing that for points $\bfx, \bfy\in \R^n$, the Euclidean norm $|\bfy - \bfx|$ is interpreted as the distance betwen $\bfx$ and $\bfy$. For example

   • Let $\bfa$ and $\bfb$ be points in $\R^n$. Let $S$ be the set of every point in $\R^n$ whose distance from $\bfa$ is exactly twice its distance from $\bfb$. Write down a formula for $S$ (using mathematical symbols rather than words).

3. Given a $2\times n$ matrix (for $n\ge 2$), you should be able to see by inspection whether the two rows are linearly independent. Similarly with the two columns of a $n\times 2$ matrix.

4. You should be able to answer, more or less immediately, questions of the form

   Describe the set $\{ \bfx\in \R^n : A\bfx = {\bf 0} \}$ or Describe the set $\{ A\bfx : \bfx \in \R^n \}$ when $A$ is a specific $m\times n$ matrix, in particular when either $m=2$ or $n=2$. The answer should have the form This set is a $*$-dimensional subspace of $\R^*$.
   
   In fact, you should also be able to answer questions of this form when $A$ is a $1\times n$ matrix or a $m\times 1$ matrix (that is, a row vector or a column vector.)

5. You should be able to compute the dot product of two vectors in $\R^n$, and the cross product of two vectors in $\R^3$.
   
   You should also know basic properties of the dot product and cross product, and be able to use them to solve problems. These properties include

   • Cauchy's inequality, triangle inequality
   • $\bfa\cdot\bfb = 0$ iff $\bfa$ and $\bfb$ are orthogonal. (This is the definition of orthogonal).
   • $\bfa\times\bfb$ is orthogonal to both $\bfa$ and $\bfb$.

6. You should be able to translate between different visual representations of functions of two variables, such as graphs and level sets, and also to match formulas for functions with either graphs or level sets. See some of the problems below for examples.

Some practice problems

Basic skills

1. We hesitate to mention this here, because you have so completely mastered this kind of thing years ago, but in case you need a reminder: Sketch the set $$ \{ (x,y)\in \R^2 : 4x^2 - \frac 14 y^2 = 1 \}. $$ Or for a little more of a challenge, $$ \{ (x,y)\in \R^2 : 4(x-1)^2 - \frac 14 (y+1)^2 = 1 \}. $$ You can easily make up more problems of this sort, by changing numbers and/or signs. You can check your answers with Wolfram Alpha for example.

2. Let $\bfa$ be a point in $\R^n$. Let $S$ be the set of every point whose distance from $\bfa$ is exactly three times its distance from the origin.

   • Write down a mathematical expression for $S$.
   • Prove that $S$ is actually a sphere of radius $r$, with centre ${\bf p}\in \R^n$. Find $r$ and ${\bf p}$. (This is harder, and you should only try it if you think it is interesting.)

3. As quickly as possible: Let $$ A := \left(\begin{array}{ccc}1&2&3 \\ 3&1&2 \end{array} \right)\qquad\mbox{ and let }\quad B := A^T $$ where ${}^T$ denotes transpose. Describe the sets $$ S_1 := \{ \bfx\in \R^3 : A\bfx = {\bf 0} \}, \qquad\qquad S_2 := \{ B\bfx : \bfx \in \R^2 \}, $$ (Your answers should have the form This set is a $*$-dimensional subspace of $\R^*$.) Again, you can easily make up more problems of this character if you like.

4. Let $\bfa$ and $\bfu$ be vectors in $\R^n$, and assume $|\bfu|=1$.

   • Show that there is exactly one number $t$ such that $\bfa - t\, \bfu$ is orthogonal to $\bfu$. What is $t$?
   • Suppose we want to find a number $t$ such that $\bfa - t\, \bfu$ is orthogonal to $\bf a$. What condition must $\bfa$ and $\bfu$ satisfy in order for this to be possible?

5. In the figures below, match the contour plots on the left with the graphs on the right. Keep in mind that graphing software often does not perform very well at points where a function is discontinuous.
   Also, in the first figure, the five functions shown are $f(x,y) = (x^2+y^2)^{p/2}$ for $p = 1/2, 1, 2, 4, 8$. Match the functions with the images.

Properties of the dot product and cross product

Students shoud not feel obliged to solve all of these, but we recommend that you try at least a few of them. The ones to try last are questions 4 and 8.

1. Prove Cauchy's inequality in special case when both $\bfa$ and $\bfb$ are unit vectors in $\R^n$, that is, that $|\bfa|^2 = |\bfb|^2 = 1$.
   Hint: Start from the obvious fact that $|\bfa-\bfb|^2 \ge 0$, and deduce what you can.

2. Prove that for any vectors $\bfa, \bfb\in \R^3$, \begin{equation}\label{cd1} \bfa\cdot( \bfa\times\bfb ) = 0. \end{equation} Hint: just write it out.

3. Deduce from \eqref{cd1} above, without redoing the proof, that $\bfb\cdot( \bfa\times\bfb ) = {\bf 0} $.

4. Given three vectors $\bfa, \bfb, {\bf c}\in \R^3$, prove that $$ \bfa \cdot (\bfb \times {\bf c}) = \det( \bfa, \bfb, {\bf c}), $$ where $( \bfa, \bfb, {\bf c})$ denotes the matrix whose three columns are $\bfa, \bfb$, and ${\bf c}$, in that order.
   Hint: just write out both sides. It's a little ugly but manageable, if you are patient, and it's a good reminder of how to compute determinants.

5. Let ${\bf r}_1$ and ${\bf r}_2$ be two linearly independent row vectors, and let $A$ be the $2\times 3$ matrix whose rows are ${\bf r}_1$ and ${\bf r}_2$. Also, let ${\bf v} = {\bf r}_1 \times {\bf r}_2$. It is a fact that $$ \{ \bfx\in \R^3 : A \bfx = 0 \} = \{ t {\bf v} : t\in \R\}. $$ Why it this true? (it is fine to give an explanation in words, without a detailed mathematical argument.)

6. (Folland, question 6 on p. 9). Show that $$ | \,|\bfa| - |\bfb|\, | \le |\bfa -\bfb| \qquad\mbox{ for all }\bfa, \bfb\in \R^n. $$ Hint: start by showing that $|\bfa| - |\bfb|\, \le |\bfa -\bfb|$ for all $\bfa, \bfb$.

7. (Folland, part of question 7 on p. 9). Suppose that $\bfa,\bfb, {\bf c}\in \R^n$.

   • Show that if $\bfa\cdot \bfb = 0$ and $\bfa\times \bfb = {\bf 0}$ then either $\bfa={\bf0}$ or $\bfb = {\bf 0}$.
     Hint: use the fact that that \begin{equation}\label{cs} |\bfa\times\bfb|^2 + (\bfa\cdot \bfb)^2 = |\bfa|^2 |\bfb|^2 \qquad\mbox{ for all }\bfa, \bfb. \end{equation} This was mentioned in Section 0.2.
   • Show that if ${\bf c}\ne {\bf 0}$ and if $$ \bfa \cdot{\bf c} = \bfb \cdot{\bf c} \quad\mbox{ and }\quad \bfa \times{\bf c} = \bfb \times{\bf c} $$ then $\bfa = \bfb$.

8. For a challenge, prove \eqref{cs}.