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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due January 31
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf MthSc 434 $|$
    Advanced Engineering Mathematics $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}


\item Let $T\colon V\to W$ be a linear map between vector
  spaces. Prove that $\ker(T):=\{v\in V\mid T(v)=0\}$ is a subspace of
  $V$.


\item Let $V=\mathcal{C}^1$, the vector space of differentiable
  real-valued functions. Consider the linear operator
  $T=\frac{d}{dt}+3$.
\begin{enumerate}
  \item The kernel of $T$ can be characterized precisely by the set of
    all funtions that solve a particular differential equation. Write
    down this equation.
  \item Find the general solution for the differential equation you
    found in Part~(a), and hence an explicit formula for $\ker(T)$.
  \item Write down an explicit basis for the solution space,
    $\ker(T)$. What is the dimension of this vector space?
\end{enumerate}


\item Consider the line $x_2+3x_1=6$ in $\R^2$. In this problem, we
  will \emph{parametrize} this line, by writing it as
  $\ell(t)=t\v+\w$, where $t\in(-\infty,\infty)$ and $\v,\w\in\R^2$
  are fixed.
\begin{enumerate}
  \item Find a vector $\w=(w_1,w_2)\in\R^2$ that lies on $\ell$
    (\emph{Any} particular vector will do!)
  \item Set $(y_1,y_2)=(x_1-w_1,x_2-w_2)$, where $(w_1,w_2)$ is the
    vector you found in Part~(a). Solve for $x_1$ and $x_2$ in terms
    of $y_1$ and $y_2$, and plug them back into the orignal equation
    to get another equation for a line. This is called a \emph{change of
      coordinates}, or \emph{change of variables}. Note that this line
    goes through the origin in the $(y_1,y_2)$-coordinate system,
    i.e., it contains the vector $\mathbf{0}=(0,0)$.
    \item Write down a parameterization for the line in the
      $(y_1,y_2)$-coordinate system. Your answer will be of the form
      $m(t)=t\v$, where $\v$ is \emph{any} non-zero vector on the
      line.
  \item Write down a parameterization for the line in the
      $(x_1,x_2)$-coordinate system.
  \item Sketch both lines, $\ell(t)=t\v+\w$ and $m(t)=t\v$, on the
    same set of axes.
\end{enumerate}


\item Consider the differential equation $y'+3y=6$. 
\begin{enumerate}
  \item Find a particular solution $y_p(t)$ to this equation by
    inspection. \emph{Hint:} Try $y_p(t)=c$, where $c$ is a constant. 
  \item Set $y_h(t)=y(t)-y_p(t)$, where $y_p(t)$ is the function you
    found in Part~(a). Plug this back into the original equation to
    get an ODE for $y_h(t)$. This is called a \emph{homogeneous} ODE
    because $y_h(t)=0$ is a solution.
  \item Solve for $y_h(t)$. Compare your answer to Part~(c) of the
    previous problem.
  \item Solve for $y(t)$. Compare your answer to Part~(d) of the
    previous problem.
\end{enumerate}



\item Consider the plane $P$ in $\R^3$ given by the equation
  $x_3-4x_1=5$. In this problem, we will parametrize $P$ by writing it
  as $P(s,t)=s\v_1+t\v_2+\w$, where $s,t\in(-\infty,\infty)$ and
  $\v_1,\v_2,\w\in\R^3$ are fixed.
  \begin{enumerate}
    \item Find a vector $\w=(w_1,w_2,w_3)$ on $P$. (Any particular
      vector will do.)
    \item Set $(y_1,y_2,y_3)=(x_1-w_1,x_2-w_2,x_3-w_3)$. Change
      coordinates by plugging these values back into the equation for $P$. 
    \item Parametrize the plane in this new coordinate system by
      writing $Q(s,t)=s\v_1+t\v_2$, where $\v_1$ and $\v_2$ are any
      two vectors on the plane that are not co-linear. Note that this
      plane goes through the origin in the $(y_1,y_2,y_3)$-coordinate
      system.
    \item Write down a parameterization for the plane in the
      $(x_1,x_2,x_3)$-coordinate system.
  \item Sketch both planes, $P(s,t)=s\v_1+t\v_2+\w$ and
    $Q(s,t)=s\v_1+t\v_2$, on the same set of axes.
  \end{enumerate}


\item Let $\v=(3,4)\in\R^2$. 
  \begin{enumerate}
  \item Compute $||\v||:=\sqrt{\v\cdot\v}$.
  \item Recall that $\{\e_1=(1,0),\e_2=(0,1)\}$ is an
    \emph{orthonormal basis} for $\R^2$. Decompose $\v$ into this
    basis, i.e., write $\v=a_1\e_1+a_2\e_2$ for some
    $a_1,a_2\in\R$. 
  \item Sketch $\e_1$, $\e_2$, and $\v$ in $\R^2$. Graphically show
    what $a_1$ and $a_2$ represent in terms of the projection of $\v$
    onto the unit vectors $\e_1$ and $\e_2$.
  \item The set $\{\v_1=(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}),
    \v_2=(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})\}$ is also an
    orthonormal basis. Decompose $\v$ into this basis, i.e., write
    $\v=b_1\v_1+b_2\v_2$ for some $b_1,b_2\in\R$.
  \item On a new set of axes, sketch $\v_1$, $\v_2$, and $\v$ in
    $\R^2$. Graphically show what $b_1$ and $b_2$ represent in terms
    of the projection of $\v$ onto the unit vectors $\v_1$ and $\v_2$.
  \end{enumerate}



\item The following differential equations are linear but
  inhomogeneous. Carry out the following steps for each.
  \begin{itemize}
  \item[i.] Find a particular solution, $y_p(t)$.
  \item[ii.] Make a change of variables $y_h(t)=y(t)-y_p(t)$, and plug
    this back in to get a linear, homogeneous ODE in terms of $y_h$.
  \item[iii.] The general solution is now $\ker(T)$, for some linear
    operation $T$. Find $T$.    
  \item[iv.]  Solve for $y_h(t)$, and use this to find $y(t)$.
  \end{itemize}
  \begin{enumerate}
  \item $y'+3y=6$
  \item $y'+3y=2t+4$
  \item $y'+3y=2e^{4t}$
  \item $y'+3y=\cos 2t$
  \item $y''+4y=e^{t}$
  \end{enumerate}

\end{enumerate}

\end{document}
