\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\v{\boldsymbol{v}}
\def\bb{\boldsymbol{b}}
\def\be{\boldsymbol{e}}
\def\A{{\boldsymbol A}}
\def\B{{\boldsymbol B}}
\def\I{{\boldsymbol I}}

\begin{document}

\title{Finite Mathematics}
\author{Mock Midterm Exam}
\date{March 24, 2020}
\maketitle

\begin{enumerate}



\item Find a parametric description of the plane which goes through
the points
\[
  \boldsymbol a =
  \begin{pmatrix}
    0 \\ 1 \\ 2
  \end{pmatrix} \,, \qquad
  \boldsymbol b =
  \begin{pmatrix}
    -3 \\ 2 \\ -1
  \end{pmatrix} \,, \qquad
  \boldsymbol c =
  \begin{pmatrix}
    1 \\ 0 \\ 1
  \end{pmatrix} \,.
\]
\hfill(5)

\item Mark the following statements as \emph{always true},
\emph{sometimes true}, or \emph{never true}.
\begin{enumerate}
\item If the augmented matrices of two systems of linear equations are
row equivalent (i.e., can be transformed into each other by elementary
row operations), then both systems have the same solution set.

\item To multiply one row of the augmented matrix by a constant is an
elementary row operation.

\item Any linear system with $n$ equations and $n$ unknowns can have
at most $n$ solutions.

\item If a system of linear equations has two different solutions,
then it has infinitely many solutions.

\item A homogeneous system of linear equations can be inconsistent.

\item A homogeneous system of linear equations has infinitely many
solutions.

\item $A \bx = \bb$ has infinitely many solutions if the columns of $A$
are linearly dependent.

\item If $A$ is a $4\times 10$ matrix, the system $A \bx = \bb$ has a
unique solution.

\item If $A$ is a $10 \times 4$ matrix, the system $A \bx = \bb$ has a
unique solution.

\item If $A$ is an invertible square matrix, the system $A \bx = \bb$
has a unique solution.
\end{enumerate}
\hfill(10)


\item
Let
\[
  A =
  \begin{pmatrix}
    1 & 3 & 0 & -4 \\
    2 & 6 & 0 & -4  
  \end{pmatrix} \,,
  \qquad
  \bb = 
  \begin{pmatrix}
    1 \\ 1 
  \end{pmatrix} \,.
\]

\begin{enumerate}
\item Find the general solution to the linear system $A \bx = \bb$.
\item Find the least-norm solution to the linear system $A \bx = \bb$.
\end{enumerate}
\hfill (5+5)


\item Which of the pairs of the following vectors are perpendicular,
which ones are not?
\[
  \boldsymbol u =
  \begin{pmatrix}
    -1 \\ 3 \\ 1
  \end{pmatrix} \,, \qquad
  \boldsymbol v =
  \begin{pmatrix}
    1 \\ 0 \\ 1
  \end{pmatrix} \,, \qquad
  \boldsymbol w =
  \begin{pmatrix}
    0 \\ 4 \\ 0
  \end{pmatrix} 
\]
\hfill (5)


\item Consider the plane with parametric representation
\[
  \bx =
  \begin{pmatrix}
    -1 \\ 1 \\ 1
  \end{pmatrix}
  + \lambda
  \begin{pmatrix}
    1 \\ -1 \\ 2
  \end{pmatrix}
  + \mu
  \begin{pmatrix}
    3 \\ 0 \\ -1
  \end{pmatrix} \,.
\]
Given the points
\[
  \boldsymbol p =
  \begin{pmatrix}
    -5 \\ -1 \\ 7
  \end{pmatrix} \,,
  \qquad \boldsymbol q =
  \begin{pmatrix}
    3 \\ 32 \\ 6
  \end{pmatrix} \,,
\]
show that one of the points $\boldsymbol p$ or $\boldsymbol q$ lies in
the plane, the other one does not.

In each case, compute the values of the parameters $\lambda$ and $\mu$
which correspond to the given point in the plane, resp.\ to the point
in the plane closest to the given point.
\hfill (15)

\item Show that if $A$ has linearly independent columns, then $A^T A$
is invertible.  \hfill(5)



\end{enumerate}
\end{document}
