---- > [!definition] Definition. ([[multilinear map]]) > For $M_{1},\dots,M_{_{\ell}},P$ [[module|modules]] over a [[ring]] $R$, a map $\varphi: M_{1} \times \cdots \times M_{\ell} \to P$ is called **$R$-multilinear** if it is $R$-[[linear in the ith variable|linear in each factor]]. In the special case $P=R$, $\varphi$ is called a **multilinear $\ell$-form**, or **(covariant) $k$-tensor**. > The set of $R$-multilinear maps $M_{1} \times \dots \times M_{\ell} \to P$ has a clear $R$-[[module]] structure; we denote that module by $L(M_{1},\dots,M_{\ell}; P)$. We in particular denote $L(V_{},\dots,V, \mathbb{F})$ by $\text{Mult}_{\mathbb{F}}^{\ell}(V)$ or just $\text{Mult}^{\ell}(V)$ for $V$ an finite-dimensional [[vector space]] over $\mathbb{F}$. > [!basicnonexample]+ > $f(\v x, \v y)=x_{1}+2x_{2}+3y_{1}+4y_{2}$ is *not* a $2$-[[multilinear map|tensor]] on $\rr^{2}$: $f(0x,y)=3y_{1}+4y_{2} \neq 0(f(\v x, \v y))$. > [!basicexample]- > > An important special case (enough to get its own note) is the [[bilinear map]]. > > $f(\v x, \v y) = x_{1}y_{1} + 2x_{1}y_{2}+3x_{2}y_{1}+4x_{2}y_{2}$ is a $2$-[[multilinear map]] on $\rr^{2}$. > [!basicexample] The General $1$-tensor and $2$-tensor on $\rrn$. > The general $1$-tensor is an element of the [[dual vector space|dual vector space]] of $\rrn$; we know this is [[linear isomorphism|linearly isomorphic]] to $\rrn$ [[module is free iff admits basis|choosing a]] [[basis]], so it follows that we can identify $1$-[[multilinear map]]s on $\rrn$ with row (or column) vectors. > \ > Next, let $f \in \text{Mult}^{2}(\rrn)$. Let $\v a_{1},\dots,\v a_{n}$ be an arbitrary [[basis]] of $\rrn$. > > Using [[basis of mult k V]], we can represent $f$ using elementary $2$-tensors, comprised of [[dual basis]] elements: $\begin{align} > \textcolor{Thistle}{f}(\v x, \v y)= & \textcolor{Thistle}{\sum_{(i,j) \in \nn_{n}^{2}}^{} c_{ij}(\phi_{i} \otimes \phi_{j})}(\v x, \v y) \\ > = & \sum_{(i,j) \in \nn_{n}^{2}}^{} c_{ij}\phi_{i}(\v x) \phi_{j}(\v y) \\ > = & \sum_{i,j=1}^{n} c_{ij} \phi_{i}(x_{1} \v a_{1} + \dots + x_{n} \v a_{n})\phi_{j}(y_{1} \v a_{1} + \dots + y_{n} \v a_{n}) \\ > = & \sum_{i,j=1}^{n} c_{ij} x_{i}\phi_{i}(\v a_{i}) y_{j} \phi_{j}(\v a_{j}) \\ > = & \sum_{i,j=1}^{n} c_{ij} x_{i}y_{j} \\ > = & \v x^{\top}C \v y > \end{align}$ > for some $C \in \rr ^{n \times n}$. For more, see [[tensor of type (p,q)|tensor]]. -- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```