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\begin{document}

\thispagestyle{fancy}
\lhead{Advanced Geometry 2017--2018}
\rhead{ENS de Lyon}
\begin{center}
\Large{Multilinear algebra}
\end{center}
\vspace{-7mm}
\rule{\linewidth}{0.5pt}


\begin{exo}[Tensor products over $\R$]
Let $E$ and $F$ be two $\R$-vector spaces of dimension $n$ and~$m$, respectively. We denote by $e=(e_j)$ a basis of $E$ and by $f=(f_i)$ a basis of $F$.
\begin{enumerate}
\item Let $\alpha \in (E^*)^{\otimes k}$, identify the coordinates of $\alpha$ in the basis of $(E^*)^{\otimes k}$ associated with $e$.
\item Give a natural isomorphism between $E^*\otimes F$ and $\mathcal{L}(E,F)$.
\item Let $L : E\to F$ be a linear map whose matrix is $M=(m_{j}^i)$ in the bases $e$ and $f$. 
We define $L^*: \alpha \mapsto \alpha \circ L$ from $F^*$ to $E^*$. Give the matrix of $L^*$ in the dual bases $e^*$ and $f^*$.
\end{enumerate}
\end{exo}

\begin{exo}[Exterior product and determinant]
Let $E$ be an $n$-dimensional vector space. We denote by $(e_i)$ a basis of $E$ and by $(e^i)$ its dual basis. Let $k \in \{1,\dots,n\}$ an integer.
\begin{enumerate}
\item Let $\alpha$ and $\beta$ be multilinear forms, check that $\alt(\alpha \otimes \beta) = \alt( \alt(\alpha) \otimes \alt(\beta))$.
\item Let $\alpha^1,\dots , \alpha^k \in E^*$, show that $\left(\alpha^1 \wedge \dots \wedge \alpha^k\right)(v_1,\dots,v_k)=\det\left((\alpha^i(v_j))_{1\leq i,j\leq k}\right)$, for any $v_1,\dots,v_k \in E$.
\item For any $I \subset \{1,\dots,n\}$, we denote by $\card(I)$ its cardinal and by $\norm{I} = \sum_{i \in I} i$ its length. Using the associativity and anticommutativity of $\wedge$ and the previous question, prove the following ``Laplace expansion'': for any matrix $M$ of dimension $n$,
\begin{equation*}
\det(M) = \sum_{\substack{I,J \subset \{1,\dots,n\}\\ \card(I)=k=\card(J)}} (-1)^{\norm{I} + \norm{J}} \det(M_{I,J}) \det(M_{I^C,J^C}),
\end{equation*}
where $I^C$ (resp.~$J^C$) is the complementary set of $I$ (resp.~$J$) and $M_{I,J}$ is the submatrix of $M$ formed by the coefficients which indices lie in $I \times J$. What happens when $k=1$?
\end{enumerate}
\end{exo}

\begin{exo}[Pullback]
Let $E$ and $F$ be two vector spaces and $L:E \to F$ be a linear map.
\begin{enumerate}
\item For any alternating forms $\alpha$ and $\beta$, show that $L^*(\alpha \wedge \beta) = L^*(\alpha) \wedge L^*(\beta)$.
\item  Let $(e_j)$ and $(f_i)$ be bases of $E$ and $F$ respectively. We denote by $M=(m_{j}^i)$ the matrix of $L$ in these bases. Let $J =\{j_1,\dots,j_k\}\subset \{1\dots,n\}$ be such that $1\leq j_1<\dots < j_k \leq n$, we denote $e^J = e^{j_1}\wedge \dots \wedge e^{j_k}$ and use similar notations for $F$. Let $\omega=\sum \omega_I f^I$, where we sum over subsets $I \subset \{1,\dots,n\}$ of cardinal $k$. Express $L^*(\omega)$ in the basis $(e^J)$.
\end{enumerate}
\end{exo}

\begin{exo}[Exterior algebra]
\begin{enumerate}
\item Is there an alternating multilinear form $\alpha$ on a vector space $E$ such that $\alpha \wedge \alpha \neq 0$?
\item Is there a non-zero alternating form commuting with any other?
\end{enumerate}
\end{exo}

\begin{exo}[Decomposable forms]
Let $E$ be a vector space of dimension $n$. An alternating $k$-linear form on $E$ is said to be \emph{decomposable} if it can be written as the alternating product of $k$ linear forms. If not, it is said \emph{indecomposable}.
\begin{enumerate}
\item Show that linear forms and alternating $n$-linear forms are always decomposable.
\item Let $\alpha \in E^*\setminus\{0\}$, show that an alternating $k$-linear form $\omega \neq 0$ is divisible by $\alpha$ (that is, can be written as $\alpha \wedge \beta$) if and only if $\alpha \wedge \omega = 0$.
\item Let $(\alpha,\beta,\gamma,\delta)$ be linearly independent in $E^*$. Is the $2$-form $ \omega = \alpha \wedge \beta + \gamma \wedge \delta$ decomposable?
\item Is an $(n-1)$-form $\omega$ always decomposable (assuming $n>1$)? Consider $ \phi_\omega : \alpha \mapsto \alpha \wedge \omega$ from $E^*$ to $\bigwedge^n E^*$.
\end{enumerate}
\end{exo}

\end{document}