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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due July 3 (Monday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2017 (online) $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: Linear maps, inner products, and orthogonality}
\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 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
  $V$. \bigskip


\item Let $V=\mathcal{C}^1(\R)$, 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? \bigskip
\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 $\v_2$. \bigskip
  \end{enumerate}



\item For this problem, consider the vector space $V=\R^3$ and use the
  vector dot product as the inner product.
  \begin{enumerate}
  \item Show that the set of vectors $\{\v_1,\v_2,\v_3\}$, where 
    \[
    \v_1=(1,2,-2)\,,\qquad \v_2=(0,1,1)\,,\qquad \v_1=(-4,1,-1)\,.
    \]
    is an orthogonal set, but not orthonormal. 
  \item Normalize $\v_1$, $\v_2$, and $\v_3$ to get an
    \emph{orthonormal} basis of $\R^3$. That is, compute the
    following:
    \[
    \mathcal{B}=\{\n_1,\n_2,\n_3\}\,\qquad\mbox{where}\quad
    \n_i=\frac{\v_i}{||\v_i||}\,.
    \]
  \item Use the dot product to express the vector $\w=(1,2,3)$ in terms
    of $\n_1$, $\n_2$, and $\n_3$. That is, find $C_1$, $C_2$, and $C_3$
    such that
    \[
    \w=C_1\n_1+C_2\n_2+C_3\n_3\,.
    \]
  \end{enumerate}



\item Let $\R_3[x]=\{a_3x^3+a_2x^2+a_1x+a_0\mid a_i\in\R\}$, the
  vector space of polynomials of degree at most $3$. Define the
  following \emph{inner product} on $\R_3[x]$:
  \[
  \<f,g\>=\int_{-1}^1 f(x)g(x)\,dx\,.
  \]
  \begin{enumerate}
    \item Verify that this is indeed an inner product on $\R_3[x]$.
    \item Consider the two sets
      \[
      \mathcal{B}_1=\{1,x,3x^2-1,5x^3-3x\}\,,\qquad
      \mathcal{B}_2=\{1,x,x^2,x^3\}
      \]
      that are both bases for $\R_3[x]$. Show that $\mathcal{B}_1$
      is an orthogonal set, but $\mathcal{B}_2$ is not. (The set
      $\mathcal{B}_1$ are the first four \emph{Legendre polynomials},
      $P_n(x)$ for $n=0,\dots,3$. When we study Sturm-Liouville
      theory, we will see why the Legendre polynomials are always orthogonal!)
    \item For each $f\in\mathcal{B}_1$, compute the \emph{norm} of
      $f$, which is defined as $||f||=\<f,f\>^{1/2}$. Find an
      \emph{orthonormal} basis for $\R_3[x]$ by normalizing the
      elements in $\mathcal{B}_1$.
    \item Consider the polynomial $f(x)=3x^3-2x^2+4$. Use
      orthogonality to write $f(x)$ using the elements in
      $\mathcal{B}_1$. That is, find $C_0$, $C_1$, $C_2$, and $C_3$
      such that
      \[
      3x^3-2x^2+4=C_0+C_1x+C_2(3x^2-1)+C_3(5x^3-3x).
      \]
  \end{enumerate}



\end{enumerate}

\end{document}
