Multilinear Forms#
Definition: Linear Form
Let \(V\) be a vector space over a field \(F\).
A \(k\)-linear form of over \(V\) is a multilinear transformation of the following form:
\[f: V^k \to F\]
For \(k=1\) we also say that \(f\) is just a linear form and for \(k=2\) we say that \(f\) is a bilinear form. In general, we call \(f\) a multilinear form.
Example: \(\mathbf{v}^{\mathsf{T}} M \mathbf{w}\)
Any function \(f: F^{n} \times F^{n} \to F\) defined as
\[ f(\mathbf{v}, \mathbf{w}) = \mathbf{v}^{\mathsf{T}}M\mathbf{w} \]
for some symmetric square matrix \(M \in F^{n\times n}\) is a bilinear form.
Example: \(\int_0^1 f(x) g(x) \mathop{\mathrm{d}x}\)
The Riemann-integral \(\int_0^1 f(x) g(x) \mathop{\mathrm{d}x}\) is a bilinear form on the vector space of all continuous real functions on \([0;1]\).