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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due October 21
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2016 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 7, pages 77--100. \\

\begin{enumerate}


\item This problem is about \emph{rational canonical
  form}. Consider the following matrix:
  \[
  M=\begin{bmatrix}0 & 0 & \cdots & 0 & -a_0 \\ 
  1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ 
  \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -a_{n-1} \\ 
  \end{bmatrix}
%  M_n=\begin{bmatrix}0&-a_0\\I_{n-1}&-\textbf{a}_{n-1}\end{bmatrix}
%  \quad\text{where}\;\;
%  \textbf{a}_{n-1}=\begin{bmatrix}a_1\\a_2\\\vdots\\a_{n-1}\end{bmatrix}.
  \]
  \begin{enumerate}
  \item Show that the minimal polynomial of $M$ is
    \[
    f(s)=s^n+a_{n-1}s^{n-1}+\dots+a_1s+a_0\,. 
    \]
  \item Let $X$ be a 4 dimensional vector space over $\R$ with
    basis $\{x_1,x_2,x_3,x_4\}$ and let $T:X\rightarrow X$ be a linear
    map such that
    \[
    T(x_1)=x_2,\quad T(x_2)=x_3,\quad T(x_3)=x_4,\quad
    T(x_4)=-x_1-4x_2-6x_3-4x_4. 
    \]
  Is $T$ diagonalizable over $\C$? Why or why not? \\
  \end{enumerate}



\item Prove that $||x||=\max\big\{(x,y):y\in K^n\mbox{ with }||y||=1\big\}$. \\



\item Let $f$ and $g$ be continuous functions on the interval $[0,1]$.
  Prove the following inequalities.
  \begin{enumerate}
  \item $\displaystyle{\left(\int_0^1f(t)g(t)\,dt\right)^2\leq
    \int_0^1f(t)^2\,dt\int_0^1g(t)^2\,dt}$
  \item $\displaystyle{\left(\int_0^1(f(t)+g(t))^2\,dt\right)^{1/2}\leq
    \left(\int_0^1f(t)^2\,dt\right)^{1/2}+\left(\int_0^1g(t)^2dt\right)^{1/2}}$. \\
  \end{enumerate}



\item Use the Gram-Schmidt process to find an orthonormal basis for
  the subspace of $\R^4$ spanned by $y_1=(1,2,1,1)$, $y_2=(1,-1,0,2)$
  and $y_3=(2,0,1,1)$. \\



  % Hiren HW 10.3 
\item Let $X$ be the vector space of all continuous real-valued
  functions on $[0,1]$. Define an inner product on $X$ by
  \[
  (f,g)=\int_0^1 f(t)g(t)\,dt\,.
  \]
  Let $Y$ be the subspace of $X$ spanned by $f_0,f_1,f_2,f_3$, where
  $f_k(x)=x^k$. Find an orthonormal basis for $Y$. 


\end{enumerate}

\end{document}







  
