## Equation for a Real Valued Linear Function on Rn

Theorem
If
$\omega_{\mathbf{x}}$
is a real valued function on
$\mathbb{R}^n$
then
$\omega_{\mathbf{x}}(\mathbf{a}) = \sum_{i=1}^n f_i (\mathbf{x}_0 ) dx_i (\mathbf{a})$
for all
$\mathbf{a} \in \mathbb{R}^n$
where
$f_i (\mathbf{x}_0)$
are real numbers.
Proof
$\omega (\mathbf{x})$
is a real valued linear function on
$\mathbb{R}^n$

Any vector
$\mathbf{a} \in \mathbb{R}^n$
can be written
$\mathbf{a}=a_1 \mathbf{e}_1 + ... + a_n \mathbf{e}_n$
.
The
$a_i$
are components of
$\mathbf{a}$
and the
$\mathbf{e}_i$
are the base vectors.
Since
$\omega_{\mathbf{x}}$
is a real valued function,
\begin{aligned} \omega_{\mathbf{x}_0}(\mathbf{a}) &= \omega_{\mathbf{x}_0}(a_1 \mathbf{e}_1 + ... + a_n \mathbf{e}_n) \\ &=a_1 \omega_{\mathbf{x}_0} (\mathbf{e}_1) +...+ a_n\omega_{\mathbf{x}_0} (\mathbf{e}_n) \end{aligned}

For a fixed
$x_0$
all the
$\omega_{\mathbf{x}_0} (\mathbf{e}_i)$
are real numbers hence
$\omega_{\mathbf{x}}(\mathbf{a}) = sum_{i=1}^n f_i (\mathbf{x}_0 ) dx+i (\mathbf{a})$
for all
$\mathbf{a} \in \mathbb{R}^n$
then
$f_i (\mathbf{x}_0)$