\[V\]
be a vector space over a field \[F\]
, let \[r\]
be a positive integer.and let \[L\]
be a multilinear function (linear in each argument) \[L:V^r \rightarrow F\]
.Let
\[A= \left( \begin{array}{cccc} a_{11} & a_{12}& \ldots & a_{1n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{r1} & a_{r2} & \ldots & a_{rn} \end{array} \right)\]
be the matrix with rows taken as \[r\]
vectors in \[V\]
.Let
\[e_i\]
the the ith basis vector with a 1 in the ith position and other entries zero, then\[\alpha_i = (a_{i1}e_1, \; a_{i2}e_2 ,..+a_{in}e_n, \; i=\sum_{j_i=1}^n a_{i_j}e_j \]
.Then
\[\]
\[ L\]
is multilinear so\[\begin{equation} \begin{aligned} L(\sum_{j_1=1}^n a_{1j_1}e_j , \; \alpha_2 ,..., \alpha_r ) &= \sum a_{1j_1} L(e_j , \; \alpha_2 ,..., \alpha_r ) \\ &= \sum a_{1j_1} L(e_j , \; \sum_{j_2=1}^n a_{2j_2}e_j ,..., \alpha_r )\\ &= \sum_{j_1=1}^n a_{1j_1} \sum_{j_2=1}^n a_{2_{j_2}} L(e_j , \; e_j ,..., \alpha_r ) \end{aligned} \end{equation}\]
Continuing in this way we have
\[\begin{equation} \begin{aligned} & L(\sum_{j_1=1}^n a_{1_{j_1}}e_j , \; \sum_{j_2=1}^n a_{2_{j_2}}e_j ,..., \sum_{j_r=1}^n a_{r_{j_r}}e_j ) \\ &== \sum^n_{j_1, \; j_2, \; j_r=1} a_{1j_1}a_{2j_2}...a_{rj_r} L(e_j , \; e_k ,..., e_r ) \end{aligned} \end{equation}\]
There is one term for each
\[L(e_j , \; e_k ,..., e_r )\]
(\[n^r\]
in total), the multilinear function is determined by these values and the last expression is the general r - linear form.