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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due October 22
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2021 $|$ M.~Macauley}}


\begin{document}

$\;$\vspace{-2mm}

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

\begin{enumerate}

  
\item Consider the vector space of all polynomials in $\C[x,y]$ of
  total degree at most $2$,
  \[
  X=\Big\{\sum a_{i,j}x^iy^j\mid a_{i,j}\in\C,\;0\leq i+j\leq 2\Big\},
  \]
  and consider the linear map
  \[
  D:X\longrightarrow X,\qquad f\longmapsto f+\frac{\partial
    f}{\partial x}+\frac{\partial f}{\partial y}.
  \]
  \begin{enumerate}
  \item Write $D$ in matrix form, with respect to the ordered basis
    $1,x,y,x^2,xy,y^2$.
  \item Find the minimal and characteristic
    polynomials, and the Jordan canonical form.
  \item Find a basis of generalized eigenvectors of $D$. 
  \item Conjecture how this generalizes to polynomial of total degree
    at most $n$.
  \end{enumerate}  
  
  %%------------------------------------------------------------------------ 

\item Let $X$ be the $xy$-plane and $A\colon X\to X$ be a $45^\circ$
  counterclockwise rotation.
  \begin{enumerate}
  \item Let $v_0=e_1=(1,0)^T$, $v_1=Av_0$, and $v_2=A^2v_0$. Write
    $v_2$ as a linear combination of $v_0$ and $v_1$, and use this to
    find the minimal polynomial of $A$.
  \item Write the matrix of $A$ with respect to the basis $v_0,v_1$,
    and compare it to the Jordan canonical form.
  \item Repeat the previous parts for a linear map $A\colon X\to X$ with eigenvalues $\lambda_{1,2}=re^{\pm i\theta}$.
  \item Re-write the matrices in Part~(c) in terms of $a$ and $b$,
    where $a\pm bi=re^{\pm i\theta}$.
  \end{enumerate}

  %%------------------------------------------------------------------------   
  
\item 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 Show that if a $\deg f(x)<n$, then $f(M)\neq 0$. [\emph{Hint}:
    Show that $f(M)e_1\neq 0$.]
  \item Show that the minimal polynomial of $M$ is
    $
    f(t)=t^n+a_{n-1}t^{n-1}+\dots+a_1t+a_0. 
    $
  \end{enumerate}

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

\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 $\C$? \medskip

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

\end{document}






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





  

