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\begin{document}
\centerline{Math 201 Homework for Tuesday, Week 11}
\bigskip
Due: Tuesday, November 16.

{\sc reminder:} You have a presentation proposal due next Tuesday, November 23. Please start thinking about the topic of your presentation: an application of linear algebra to other fields.

\begin{prob}
  Let $V=\mathcal{P}_3(\R)$, the vector space over $\mathbb{R}$ consisting of all polynomials with real
  coefficients having degree at most $3$. Define the following linear transformation on $V$ (in which the prime denotes
  differentiation),
  \begin{align*}
    L\colon V&\to V\\
   f&\mapsto xf'+f'.
  \end{align*}
  \begin{enumerate}
    \item Write the matrix of $L$ with respect to the ordered basis $\langle 1, x, x^2 , x^3\rangle$
      of $V$.
    \item What are the eigenvalues of $L$?
    \item Does $V$ have a basis of eigenvectors of $L$? If so, give such a basis
      ({\bf written as polynomials} not tuples of real numbers), and if not, explain why not.
  \end{enumerate}
\end{prob}
\begin{prob}
  Consider the matrix
    \[
  B = 
  \left(\begin{array}{ccc}
    2&1&1\\
    0&3&2\\
    0&0&2
  \end{array} \right).
  \]
  \begin{enumerate}
    \item What are the algebraic and geometric multiplicities of each of the
      eigenvalues of~$B$?
    \item Explain whether~$B$ is diagonalizable in terms of the geometric
      multiplicities of its eigenvalues.
  \end{enumerate}
\end{prob}
\begin{prob}
  Consider an $n \times n$ matrix $A$ over $\C$. As mentioned in class, the characteristic polynomial of $A$ is of the form
  \[ p_A(t)=\det(A-tI_n)=(-1)^nt^n+b_{n-1}t^{n-1} + \dots + b_1t+b_0. \]
  \begin{enumerate}
   \item Prove that $b_0$ is equal to the determinant of $A$.
   \item Prove that $b_{n-1}=(-1)^{n-1}\trace(A)$, where $\trace$ denotes the trace.
  
   ({\sc hint:} Using the permutation expansion identify where the possible terms of degree $n-1$ arise from.)
   \item Prove that the product of all eigenvalues of $A$ (with multiplicity) is
     equal to $\det(A)$. (You may use the fact the characteristic polynomial
     will factor completely into linear factors,
     $p_A(t)=(-1)^{n}\prod_{i=1}^{n} (x_{i}-\lambda_{i})$, since we are working over~$\C$.)
   \item Prove that the sum of all eigenvalues of $A$ (with multiplicity) is equal to $\trace(A)$.
  \end{enumerate}
  {\sc note:} When $n=2$, this result says that 
  \[p_A(t)=t^2-\trace(A)t+\det(A).\]
\end{prob}
\end{document}



