\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\by{\boldsymbol{y}}
\def\ba{\boldsymbol{a}}
\def\be{\boldsymbol{e}}

\begin{document}

\title{Finite Mathematics}
\author{Homework 1}
\date{Due in class Wednesday, February 12, 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.\ 41, Excercises 1.2, 1.3, 1.4

\item Find a parametric description of the plane which goes through
the point
\[
\begin{pmatrix}
  1 \\ -1 \\ 0
\end{pmatrix}
\]
and the line with parametric description
\[
  \bx =
  \begin{pmatrix}
    4 \\ 0 \\ -4
  \end{pmatrix}
  + \lambda \,
  \begin{pmatrix}
    3 \\ 2 \\ 1
  \end{pmatrix} \,.
\]
(Cf.\ *Hefferon p.\ 41, Exercise 1.6)
\hfill (3)

\item VMLS, p. 25, Exercise 1.7
\hfill (2)

\item VMLS, p. 26, Exercise 1.12
\hfill (3)

\item *VMLS, p. 27, Exercise 1.16

\item *VMLS, p. 42, Exercise 2.1

% \item Recall from class that a function $f \colon \R^n \to \R$ is
% \emph{affine} if the function $g(\bx) = f(\bx) - f(0)$ is linear.  Show
% that an affine function has the \emph{restricted superposition
% property}
% \[
%   f(\alpha \bx + \beta \by) = \alpha f(\bx) + \beta f(\by)
%   \quad \text{whenever} \quad \alpha + \beta = 1 \,.
% \]

\item Recall from class that an affine function $f \colon \R^n \to \R$
can be written as $f(\bx) = \ba^T \bx + b$ for some $\ba \in \R^n$ and
$b \in \R$.  Verify that $b = f(0)$ and
\[
  \ba =
  \begin{pmatrix}
    f(\be_1) - f(0) \\ \vdots \\ f(\be_n) - f(0)
  \end{pmatrix} \,.
\]
\end{enumerate}
\hfill (2)

\end{document}
