Equation for a Real Valued Linear Function on Rn
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,\[\begin{equation} \begin{aligned} \omega_{\mathbf{x}_0}(\mathbf{a}) &= \omega_{\mathbf{x}_0}(a_1 \mathbf{e}_1 + ... + a_n \mathbf{e}_n) \\ &=a_1 \omega_{\mathbf{x}_0} (\mathbf{e}_1) +...+ a_n\omega_{\mathbf{x}_0} (\mathbf{e}_n) \end{aligned} \end{equation}\]
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 \[f_i (\mathbf{x}_0) \]
