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\lhead{{\sf Summer Bridge Course (Algebra)}: Thursday July 23}
\chead{}\rhead{\thepage}
\lfoot{Due Friday, July 24 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
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\begin{document}

%\(\;\) \vspace{-4mm}

\noindent \textbf{Topics}: Multilinearity. \\


\noindent \textbf{Do}: Answer the following questions. Throughout, assume that \(X\) is a finite-dimensional vector space over a field \(K\). A \(k\)-linear form \(f\colon X^k\to K\) is:
\begin{itemize}
    \item \emph{symmetric} if \(f(x_1,\dots,x_k)=\pi\cdot f(x_1,\dots,x_k):=f(x_{\pi^{-1}(1)},\dots,x_{\pi^{-1}(k)})\) for all  \(\pi\in S_k\),
    \item  \emph{skew-symmetric} if \(\tau\cdot f(x_1,\dots,x_k)=-f(x_1,\dots,x_k)\) for all transpositions \(\tau=(ij)\in S_k\),
    \item \emph{alternating} if \(f(x_1,\dots,x_k)=0\) whenever \(x_i=x_j\). \\
\end{itemize}

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

%% HW 13, Problem #1
  \begin{exercise}      
    Let \(f\) be a bilinear form over a vector space \(X\) with basis \(\{x_1,x_2\}\).
  \begin{enumerate}[label=(\roman*)]
  \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.
  \item Repeat Part~(a) but assume that \(f\) is symmetric, and that
    \(f(x,x)=0\) for all \(x\in X\). 
  \item Repeat Part~(a) but now assume that \(X\) is \(3\)-dimensional, with basis \(\{x_1,x_2,x_3\}\). \\ \bigskip
  \end{enumerate}
\end{exercise}

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

%% HW 13, Problem #2
 \begin{exercise}
  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 \(i^{\mathrm{th}}\) column of \(A\) to the \(j^{\mathrm{th}}\)
  column of \(A\), for some \(i\neq j\). Show that \(\det A=\det B\). You may
  assume that the determinant is an alternating \(n\)-linear form. \\
  \end{exercise}
  
 %%--------------------------------------------------------------------- 

%% HW 13, Problem #3
 \begin{exercise}
Let \(f\) be an alternating \(k\)-linear form. Show that if \(y_1,\dots,y_k\) are linearly dependent, then \(f(y_1,\dots,y_k)=0\). Then give an explicit counterexample to show how the converse fails. 
\end{exercise}

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

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