\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\bx{\boldsymbol{x}}
\def\bx{\boldsymbol{x}}
\def\v{\boldsymbol{v}}
\def\ba{\boldsymbol{a}}
\def\be{\boldsymbol{e}}
\def\A{{\boldsymbol A}}
\def\B{{\boldsymbol B}}
\def\I{{\boldsymbol I}}

\begin{document}

\title{Finite Mathematics}
\author{Homework 4}
\date{Due in class Monday, March 9, 2020}
\maketitle

\emph{Note:} Assignments marked (*) are \emph{not} for bonus credit.
They will not be graded.  Do not turn them in.  However, they will be
discussed in the tutorial and typically example solutions are
available online (in case of Hefferon's book, there is a PDF with
solutions available on the book's web site.)

\begin{enumerate}

\item *Hefferon, p.\ 119, Exercises 1.19

\item *Hefferon, p.\ 119, Exercises 1.21


\item Determine whether the members of the given set of vectors are
linearly independent.  If they are linearly dependent, find a linear
relation among them.

\begin{enumerate}
\item $\displaystyle
\v_1 =
\begin{pmatrix}
  1 \\ 1 \\ 0
\end{pmatrix} \,,
\quad
\v_2 =
\begin{pmatrix}
  0 \\ 1 \\ 1 
\end{pmatrix} \,,
\quad
\v_3 =
\begin{pmatrix}
  1 \\ 0 \\ 1
\end{pmatrix}$

\item $\displaystyle
\v_1 =
\begin{pmatrix}
  1 \\ 2 \\ -2
\end{pmatrix} \,,
\quad
\v_2 =
\begin{pmatrix}
  3 \\ 1 \\ 0
\end{pmatrix} \,,
\quad
\v_3 =
\begin{pmatrix}
  2 \\ -1 \\ 1
\end{pmatrix}\,,
\quad
\v_4 =
\begin{pmatrix}
  4 \\ 3 \\ -2
\end{pmatrix}$

\end{enumerate}
\hfill(2+2)

\item *VMLS, p.~64, Exercise 3.4

\item VMLS, p.~64, Exercise 3.9
\hfill(2)

\item Compute $\lVert \bx \rVert$, $\operatorname{rms}(\bx)$ and
$\operatorname{avg}(\bx)$ for the $16$-vector $\bx$ which has
$x_1=x_2=x_3=2$ and $x_4 = \dots = x_{16}=1$.
\hfill(2)


\item (Cf.\ VMLS, Exercise 3.16.)  Suppose $\bx$ is an $n$-vector and
$\alpha$ and $\beta$ are scalars.  Show that
\begin{enumerate}
\item $\operatorname{avg}(\alpha \bx + \beta) = \alpha
\operatorname{avg}(\bx) + \beta$,
\item $\operatorname{std}(\alpha \bx + \beta) = \lvert \alpha \rvert 
\operatorname{std}(\bx)$.
\end{enumerate}
\hfill(2)

\end{enumerate}
\end{document}
