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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due September 23
    (Wednesday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2020 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 5, pages 44--57. \medskip


\begin{enumerate}

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

\item Let $S_n$ denote the set of all permutations of $\{1,\dots,n\}$.
  \begin{enumerate}
  \item Prove that $\sgn(\pi_1\circ\pi_2)=\sgn(\pi_1)\sgn(\pi_2)$.
  \item Prove that $\sgn(\tau)=-1$  for all transpositions $\tau\in S_n$.
  \item Let $\pi\in S_n$, and suppose that
    $\pi=\tau_k\circ\cdots\circ\tau_1=\sigma_\ell\circ\cdots\circ\sigma_1$,
    where $\tau_i,\sigma_j\in S_n$ are transpositions. Prove that
    $k\equiv\ell$ mod $2$. \medskip
  \end{enumerate}

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

\item Let $f$ be a bilinear form over a $K$-vector space $X$ with basis
  $\{x_1,x_2\}$.
  \begin{enumerate}
    \item Assume $f$ is alternating. Determine a formula for $f(u,v)$
      in terms of each $f(x_i,x_j)$ and the coefficients used to express
      $u$ and $v$ with this basis. [Pun intented!]
    \item Repeat Part~(a) but assume that $f$ is symmetric and
      $f(x,x)=0$ for all $x\in X$. \medskip
  \end{enumerate}

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

\item Let $X$ be an $n$-dimensional vector space over a field $K$.
  \begin{enumerate}
  \item Prove that if $\Char K\neq 2$, then every
    skew-symmetric multilinear form is alternating.
  \item Give an example of a non-alternating skew-symmetric mulitlinear form.
  \item Give an example of a non-zero alternating multilinear form such
    that $f(x_1,\dots,x_k)=0$ for some set of linearly independent
    vectors $x_1,\dots,x_k$. \medskip
  \end{enumerate}

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

\item Let $X$ be an $n$-dimensional vector space over $\R$, and let
  $f$ be a non-degenerate symmetric bilinear form. That is, it has the
  additional property that for all nonzero $x\in X$, there is some
  $y\in X$ for which $f(x,y)\neq 0$.
  \begin{enumerate}
  \item Prove that the map $L\colon X\rightarrow X'$ given by $L\colon
    x\mapsto f(x,-)$ is an isomorphism.
  \item Show that, given any basis $x_1,\dots,x_n$ for $X$, there
    exists a basis $y_1,\dots,y_n$ such that $f(x_i,y_j)=\delta_{ij}$.
  \item Conversely, prove that if $\BBB_X=\{x_1,\dots,x_n\}$ and
    $\BBB_Y=\{y_1,\dots,y_n\}$ are sets of vectors in $X$ with
    $f(x_i,y_j)=\delta_{ij}$, then $\BBB_X$ and $\BBB_Y$ are bases for
    $X$. \medskip
  \end{enumerate} 
    
%%------------------------------------------------------------------------

\item Let $X$ be an $n$-dimensional vector space over $\R$, and let
  $f$ be a non-degenerate symmetric bilinear form. %%
  \begin{enumerate}
  \item Show that there exists $x_1\in X$ with $f(x_1,x_1)\neq 0$.
  \item Any fixed $x_1\in X$ for which $f(x_1,x_1)\neq 0$ induces a
    \emph{linear} map $T=f(x_1,-)$.  Find the dimension of the
    nullspace $Z_1:=N_T$, and show that the restriction of $f$ to
    $Z_1\times Z_1$ is again non-degenerate.
  \item Prove that $X$ has a basis $\{z_1,\dots,z_n\}$ such that
    $f(z_i,z_j)=\delta_{ij}$. \medskip
  \end{enumerate}

%%------------------------------------------------------------------------
  
\item Let $A=(c_1,\dots,c_n)$ be an $n\times n$ matrix ($c_i$ is a
  column vector), and let $B$ be the matrix obtained from $A$ by
  adding $k$ times the $i^{\rm th}$ column of $A$ to the $j^{\rm th}$
  column of $A$, for $i\neq j$. Prove that $\det A=\det B$. You may
  assume that the determinant is an alternating $n$-linear form.

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

\end{enumerate}

\end{document}







  
