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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due June 30 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2017 (online) $|$ M.~Macauley}}


\begin{document}


$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: Vector spaces, linear independence, and bases}
\end{center}

%\noindent Read the following:
%\begin{itemize}
%\item Chapters 1 \& 2 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
%\item VGT Exercises 1.1--1.4, 1.8--1.12, 2.13--2.17. \vspace{-2mm}
%\item The article \emph{Group Think} by Steven Strogatz, which
%  appeared in the NY Times in 2010. \vspace{-2mm}
%\item Francis Su's essay on good mathematical writing.
%\end{itemize} 


\begin{enumerate}

\item For each of the following sets, determine if it is a \emph{vector
  space} over $\R$. If it is, give an explicit \emph{basis} and compute its
  \emph{dimension}. If it isn't, explain why not by giving an example of
  how one of the vector space properties fails.
  \begin{enumerate}
  \item The set of points in $\R^3$ with $x=0$.
  \item The set of points in $\R^2$ with $x=y$.
  \item The set of points in $\R^3$ with $x=y$.
  \item The set of points in $\R^3$ with $z\geq 0$.
  \item The plane in $\R^3$ defined by the equation $z=2x-3y$.
  \item The set of unit vectors in $\R^2$.
  \item The set of polynomials of degree $n$.
  \item The set $\R_n[x]$ of polynomials of degree at most $n$.
  \item The set of polynomials of degree at most $n$, with only
    even-powers of $x$.
  \item The set $\Per_{2\pi}(\R)$ of piecewise continuous functions,
    i.e., $f$ such that $f(x)=f(x+2\pi)$.
  \item $\C:=\{a+bi\mid a,b\in\R\}$. \bigskip
  \end{enumerate}



\item Let $v_1,v_2,w$ be three \emph{linearly independent} vectors in
  $\R^3$. That is, they do not all lie on the same plane. For each of
  the following (infinite) set of vectors, carefully sketch it in
  $\R^3$, and determine whether or not it is a vector space (i.e., a
  \emph{subspace} of $\R^3$). Explain your reasoning.
  \begin{multicols}{2}
    \begin{enumerate}
    \item $\{C v_1\mid C\in\R\}$
    \item $\{C v_1+w\mid C\in\R\}$ 
    \item $\{C_1 v_1+C_2 v_2\mid C_1,C_2\in\R\}$
    \item $\{C_1 v_1+C_2 v_2+w\mid C_1,C_2\in\R\}$ \bigskip
    \end{enumerate}
  \end{multicols}

  

\item Find the general solution to each of the following ODEs. Then,
  decide whether or not the set of solutions form a vector
  space. Explain your reasoning. Compare your answers to the previous
  problem. Recall that the general solution has the form $y(t)=y_h(t)+y_p(t)$. 
  \begin{multicols}{2}
    \begin{enumerate}
    \item $y'-2y=0$ 
    \item $y'-2y=1$ 
    \item $y''+4y=0$
    \item $y''+4y=e^{3t}$ \bigskip
    \end{enumerate}
  \end{multicols}



%\item Complete the following sentences as concisely as possible:
%  \begin{enumerate}
%  \item Two non-zero vectors $v_1,v_2\in\R^2$ are a basis for
%    $\R^2$ if and only if\dots
%  \item Three non-zero vectors $v_1,v_2,v_3\in\R^3$ are a basis for
%    $\R^3$ if and only if\dots
%  \end{enumerate}



\item For each of the following pairs of vectors $v_1=(x_1,y_1)$ and
  $v_2=(x_2,y_2)$ in (a)--(e), carry out the following steps:
  \begin{enumerate}[(i)]
    \item The lines through $v_1$ and $v_2$ generate a grid (of
      parallelograms) on the $xy$-plane. Sketch $v_1$, $v_2$, and this
      grid.
    \item Find the area of one of the parallelgrams by computing
      the determinant of the matrix $\left[\!\!\begin{array}{cc}x_1 &
        x_2 \\ y_1 & y_2\end{array}\!\!\right]$. Is this matrix invertible?
    \item Determine whether $\{v_1,v_2\}$ is a basis of $\R^2$.
  \end{enumerate}
  \begin{multicols}{2}
    \begin{enumerate}
    \item $v_1=(1,0)$, $v_2=(0,1)$ 
    \item $v_1=(2,0)$, $v_2=(0,2)$ 
    \item $v_1=(\frac{1}{2},\frac{1}{2})$,
      $v_2=(\frac{1}{2},-\frac{1}{2})$
    \item $v_1=(1,1)$, $v_2=(1,2)$ 
    \item $v_1=(1,2)$, $v_2=(1,1)$ 
    \item $v_1=(2,-1)$, $v_2=(-4,2)$ 
    \end{enumerate}
  \end{multicols}
    Summarize your conclusions in a sentence or two. \bigskip



\item For each of the following triples of vectors
  $v_1=(x_1,y_1,z_1)$, $v_2=(x_2,y_2,z_2)$, and $v_3=(x_3,y_3,z_3)$,
  carry out the following steps:
  \begin{enumerate}[(i)]
    \item Sketch $v_1$, $v_2$, and $v_3$ in $\R^3$.
    \item Use a computer to calculate the determinant of 
      $\left[\!\!\begin{array}{ccc}x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3
      \\ z_1 & z_2 & z_3\end{array}\!\!\right]$. Is it invertible?
    \item The lines through $v_1$, $v_2$, and $v_3$ generate a
      tessellation (of parallelepipeds) in $\R^3$. What do you think
      the volume of each parallelepiped is?
    \item Describe in words (e.g., line, plane, all of $\R^3$) the
      subspace $\Span\{v_1,v_2,v_3\}$. Is $\{v_1,v_2,v_3\}$ is a basis
      of $\R^3$? 
  \end{enumerate}
  \begin{enumerate}
    \item $v_1=(1,0,0)$, $v_2=(0,1,0)$, $v_3=(0,0,1)$ 
    \item $v_1=(2,0,0)$, $v_2=(0,2,0)$, $v_3=(0,0,2)$ 
    \item $v_1=(1,0,0)$, $v_2=(0,1,1)$, $v_3=(3,1,1)$
    \item $v_1=(1,0,0)$, $v_2=(0,2,-1)$, $v_3=(1,1,1)$
  \end{enumerate}
  Summarize your conclusions in a sentence or two. \bigskip



\item For each of the following, a vector space $V$ is given, along
  with a finite set $S\subset V$. Denote the subspace of $V$
  \emph{spanned} by $S$ as $\Span(S)$. Find an explicit basis for
  $\Span(S)$ and compute its dimension.
  \begin{enumerate}
  \item $V=\R^3$, $S=\{(1,0,0),(0,1,1),(1,1,1),(3,1,1)\}$. 
  \item $V=\R^2$,
    $S=\{(1,0),(0,1),(\frac{1}{2},\frac{1}{2}),(\frac{1}{2},-\frac{1}{2})\}$.
  \item $V=\mathcal{C}^\infty(\R)$, $S=\{e^{3x},e^{-3x},\cosh 3x,\sinh 3x\}$.
  \end{enumerate}


\end{enumerate}

\end{document}
