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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due November 4
    (Wednesday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2020 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 7, pages 77--100. \vspace{2mm}

\begin{enumerate}

%%------------------------------------------------------------------------ 
  
\item This problem is about \emph{rational canonical
  form}. Consider the following matrix over $\R$:
  \[
  M=\begin{bmatrix}0 & 0 & \cdots & 0 & -a_0 \\ 
  1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ 
  \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -a_{n-1}
  \\ 
  \end{bmatrix}
  \]
  \begin{enumerate}
  \item Let $e_1,\dots,e_n$ be the standard basis. Show that if a
    polynomial $f\in\R[x]$ has degree less than $n$, then $f(M)\neq
    0$. [\emph{Hint}: Notice that $Me_i=e_{i+1}$ for $i=1,\dots,n-1$.]
  \item Show that the minimal polynomial of $M$ is
    \[
    f(t)=t^n+a_{n-1}t^{n-1}+\dots+a_1t+a_0\,. 
    \]
  \item Let $X$ be a 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
    \[
    T(x_1)=x_2,\quad T(x_2)=x_3,\quad T(x_3)=x_4,\quad
    T(x_4)=-x_1-4x_2-6x_3-4x_4. 
    \]
    Find the rational and Jordan canonical forms of $T$. Is $T$
    diagonalizable over $\C$? Why or why not? \\
  \end{enumerate}

%%------------------------------------------------------------------------   

\item Given a linear map $A\colon X\to X$, define $f\colon X\to X$ by
  $f(x,y)=x^TAy$.
  \begin{enumerate}
  \item State and prove necessary and suffcient conditions
    on $A$ for $f$ to be an inner product. 
  \item Write the inner product
    $f(x,y)=3x_1y_1-x_1y_2-x_2y_1+2x_2y_2-x_2y_3-x_3y_2+3x_3y_3$ as
    $f(x,y)=x^TAy$.
  \item Find an orthonormal basis $v_1,v_2,v_3$ of $\R^3$ so that with
    respect to this basis, $f(z,w)=z^TDw$ for some diagonal matrix
    $D$.
  \item Write a formula for $f(z,w)$ like in Part~(b), but with
    respect to this new basis. \\
\end{enumerate}

%%------------------------------------------------------------------------ 

\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} \vspace{2mm}

%%------------------------------------------------------------------------ 

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

%%------------------------------------------------------------------------ 
   
\end{enumerate}

\end{document}







  
