\documentclass[12pt,fleqn]{article}
\pagestyle{empty}
\setlength{\topmargin}{-1in}
\setlength{\oddsidemargin}{-.3in}
\setlength{\textheight}{9in}
\setlength{\textwidth}{7.2in}


\usepackage{amssymb}
\usepackage{amsmath}


\newcommand{\adj}{\mathrm{Adj}}
\newcommand{\mbf}{\mathbf}
\newcommand{\R}{\mathbb{R}}
\newcommand{\ra}{\rightarrow}
\newcommand{\proj}{\mathrm{proj}}
\newcommand{\dotprod}{\bullet}

\begin{document}

\begin{center}
{\Large TEST 1

Math 2250}
\end{center}

\begin{itemize}
\item  Show all your work and make your work neat.  You will be graded on this!
\item  You can use a calculator, but I must be able to follow your work as if you had no calculator.
\item  You will be graded on both your answer and the work done to get that answer.
\end{itemize}


\begin{enumerate}


\item  (10 points)
  Solve the following differential equations:
  \begin{enumerate}
  \item  
    \[
    x \frac{dy}{dx} - y - 2x^2y=0 \qquad y(1)=1
    \]
\vspace{2in}
  \item
    \[
    \frac{dy}{dx} = e^x - y \qquad y(0)=1
    \]
  \end{enumerate}
\newpage
				   
\item  (10 points)
  Determine if you can guarantee that the following initial value problems have a solution.
  If you can guarantee a solution, can you guarantee that the solution is unique?
  (Justify your answers!)
  \begin{enumerate}
  \item  
    \[
    \frac{dy}{dx} = \frac{x}{x-y} \qquad y(1)=2
    \]
\vspace{3in}
  \item  
    \[
    \frac{dy}{dx} = x^3\sin y + y^{2/3} \qquad y(1)=0
    \]
  \end{enumerate}	
\newpage

\item (10 points)
  Consider the linear system below:
  \[
  \begin{array}{rrrrrrr}
    x_1 & + & x_2 & - & x_3 & = 0 \\
    x_1 &  &  & + & 2x_3 & = 0 \\
    2x_1 & - & 3x_2 & + & 13x_3 & = 0
  \end{array}
  \]
  \begin{enumerate}
  \item  Write the system of equations as a matrix equation.
  \item  Write the system equations in an augmented matrix and use Gaussian elimination on this matrix
    (you should end up with a reduced echelon matrix).
  \item  For the matrix $A$ below, what is a basis for the solution space for $A$?
      (notice the similarity with the linear system).
    \[
    \left[
    \begin{array}{rrr}
      1 & 1 & -1 \\
      1 & 0 & 2 \\
      2 & -3 & 13
    \end{array}
    \right]
    \]
  \item  What is the dimension of the solution space?
  \end{enumerate}
\newpage

\item (5 points)
  Determine if the set of vectors is linearly dependent or linearly independent
  (justify your answer!)
  \[
  S = \{(1,2,3,4), (1,2,-3,12), (-2,4,5,-17), (0,1,2,4), (0,0,2,-5) \}
  \]
\vspace{2in}

\item  (5 points)
  Consider the set $S$ of vectors below.
  (Justify your answers!)
  \[
  S = \{ (4,0,1), (1,2,1), (1,-6,-2) \}
  \]
  Is the vector $(1,-2,3)$ in $\mathrm{Span}(S)$?



\end{enumerate}

\end{document}