## n - forms as Linear Transformations

A differential form is a linear operator, so if
$dx_1 \wedge dx_2 \wedge ... \wedge dx_n$
is n - form operating on vectors
$(\mathbf{v}_1 , \mathbf{v}_2 , ..., \mathbf{v}_n )$
then
\begin{aligned} & dx_1 \wedge dx_2 \wedge ... \wedge dx_n (k_1 (\mathbf{v}_1 , \mathbf{w}_2 , ..., \mathbf{v}_n ) + k_2 (\mathbf{w}_1 , \mathbf{w}_2 , ..., \mathbf{w}_n )) \\&= k_1 dx_1 \wedge dx_2 \wedge ... \wedge dx (k_1 (\mathbf{v}_1 , \mathbf{w}_2 , ..., \mathbf{v}_n ) \\ &+ k_2 dx_1 \wedge dx_2 \wedge ... \wedge dx_n (\mathbf{w}_1 , \mathbf{w}_2 , ..., \mathbf{w}_n )) \end{aligned}

This means that we can represent an n - form by a matrix.
The set of n forms is also linear for each value of n, so that if
$L_1$
and
$L_2$
are n - forms then so is
$k_1 L_1 + k_2 L_2$
where
$k_1 , \: k_2$
are scalars.