Editing Multilinear map (section)

==Multilinear functions on ''n''&times;''n'' matrices==
One can consider multilinear functions, on an {{math|''n''&times;''n''}} matrix over a [[commutative ring]] {{mvar|K}} with identity, as a function of the rows (or equivalently the columns) of the matrix.  Let {{math|''A''}} be such a matrix and {{math|''a<sub>i</sub>'', 1 ≤ ''i'' ≤ ''n''}}, be the rows of {{math|''A''}}.  Then the multilinear function {{math|''D''}} can be written as

:<math>D(A) = D(a_{1},\ldots,a_{n}),</math>

satisfying

:<math>D(a_{1},\ldots,c a_{i} + a_{i}',\ldots,a_{n}) = c D(a_{1},\ldots,a_{i},\ldots,a_{n}) + D(a_{1},\ldots,a_{i}',\ldots,a_{n}).</math>

If we let <math>\hat{e}_j</math> represent the {{mvar|j}}th row of the identity matrix, we can express each row {{math|''a<sub>i</sub>''}} as the sum

:<math>a_{i} = \sum_{j=1}^n A(i,j)\hat{e}_{j}.</math>

Using the multilinearity of {{math|''D''}} we rewrite {{math|''D''(''A'')}} as

:<math>
D(A) = D\left(\sum_{j=1}^n A(1,j)\hat{e}_{j}, a_2, \ldots, a_n\right)
       = \sum_{j=1}^n A(1,j) D(\hat{e}_{j},a_2,\ldots,a_n).
</math>

Continuing this substitution for each {{math|''a<sub>i</sub>''}} we get, for {{math|1 ≤ ''i'' ≤ ''n''}},

:<math>
D(A) = \sum_{1\le k_1 \le n} \ldots \sum_{1\le k_i \le n} \ldots \sum_{1\le k_n \le n} A(1,k_{1})A(2,k_{2})\dots A(n,k_{n}) D(\hat{e}_{k_{1}},\dots,\hat{e}_{k_{n}}).
</math>

Therefore, {{math|''D''(''A'')}} is uniquely determined by how {{mvar|D}} operates on <math>\hat{e}_{k_{1}},\dots,\hat{e}_{k_{n}}</math>.