Given two \(k\)-forms \(\alpha\) and \(\beta\),
return the inner product
\(\left\langle\alpha,\beta\right\rangle\). Here our
underlying vector space \(V\) is \(\mathcal{R}^n\).
The inner product is a symmetric bilinear form defined in two stages.
First, we specify its behaviour on decomposable \(k\)-forms
\(\alpha=\alpha_1\wedge\cdots\wedge\alpha_k\) and
\(\beta=\beta_1\wedge\cdots\wedge\beta_k\) as
$$
\left\langle\alpha,\beta\right\rangle=\det\left(
\left\langle\alpha_i,\beta_j\right\rangle_{1\leq i,j\leq n}\right)
$$
and secondly, we extend to the whole of \(\Lambda^k(V)\)
through linearity.