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\begin{document}
\title{Linear Algebra Problems}
\author{David Pierce}
\date{April 2, 2019}
\publishers{Matematik B\"ol\"um\"u, MSGS\"U\\
\url{mat.msgsu.edu.tr/~dpierce/}}

\maketitle\thispagestyle{empty}

Sources include
\begin{compactitem}
\item
  Cemal Ko\c c, \emph{Topics in Linear Algebra}
\item
  Steven Roman, \emph{Advanced Linear Algebra} (2d ed.)
\end{compactitem}
  
  \begin{problem}
    $X\mapsto\uptau_X$, where $\uptau_A(Y)=AY$,
    is an isomorphism from $M$ to $\lin{K^n,K^n}$.
  \end{problem}

\begin{problem}
  If $V$ is a finite-dimensional vector space, $\tau$ is an endomorphism of $V$
  (in other words, a linear operator on $V$),
  and $\tau$ and $\tau^2$ have the same rank,
  show that the kernel and image of $\tau$ have trivial intersection.
\end{problem}

\begin{problem}
  Supposing $\{\bm a_k\colon k\in n\}$ is a basis of $V$,
  define $\tau_{ij}$ in $\lin V$ by
  \begin{equation*}
    \tau_{ij}(\bm a_k)=
    \begin{cases}
      \bm a_k&\text{ if $k\neq i$},\\
      \bm a_i+\bm a_j&\text{ if $k=i$}.
    \end{cases}
  \end{equation*}
  \begin{compactenum}[(a)]
  \item
    Under what conditions on the scalar field is each $\tau_{ij}$ invertible?
  \item
    Under what conditions on the scalar field
    do the $\tau_{ij}$ compose a basis of $\lin V$?
  \end{compactenum}
\end{problem}

\begin{problem}
  Prove concerning determinants:
  \begin{compactenum}[(a)]
  \item
  $\det(\dots,\bm v,\dots,\bm v,\dots)=0$,  
    \item
  $\det A\cdot\Id=A\adj A$,
  \item
    $\det{\trans A}=\det A$.
  \end{compactenum}
\end{problem}

  \begin{problem}
    If an abelian group has a multiplication
    (meaning a binary operation distributing in both senses over addition),
    and this has both a left and a right identity,
    show that they are equal.
  \end{problem}

\begin{problem}
  For which $2\times2$ matrices is the minimal polynomial
  the same as the characteristic polynomial?
\end{problem}

\begin{problem}
  If $A$ is a real matrix and $A^5+A^3+A=0=A^5-A^3-A$,
  what is the minimal polynomial of $A$?
\end{problem}

\begin{problem}
  Prove that the characteristic polynomial of every real $2\times2$
  symmetric matrix is the product of linear factors.
\end{problem}

\begin{problem}
  Show that $AB$ and $BA$ have the same eigenvalues.
\end{problem}

\begin{problem}
  If $A$ is diagonalizable,
  show the same for every polynomial function $f(A)$ of $A$.
\end{problem}

\begin{problem}
  If $A$ is \textbf{nilpotent,}
  that is, some power of $A$ is the zero matrix,
  show $A$ is not diagonalizable.
\end{problem}

\begin{problem}
  Find the Jordan normal form of every $5\times 5$ nilpotent matrix.
\end{problem}

\begin{problem}
  Find the Jordan normal form of every $n\times n$ complex matrix
  of which some positive power is $\Id$.
\end{problem}

\end{document}
