<s>[INST] Using least squares method by using data points A set of six data points are given in terms of $X$ and $Y$ like $(x,y)$ or just say six coordinates data set. How can I use least square method to make linear combination of $x$ , $\sin(x)$ and $\exp(x)$ that best describe the curve that passes through all those points ? I can do it if only $x$ or $\sin(x)$ or $\exp$ is present.
 [/INST] @Claude Leibovici provides critical insights. Some details are added.
Input data: $m=6$ points
$$
 \left\{ x_{k}, y_{k} \right\}_{k=1}^{m}
$$
Linear model:
$$
 y(x) = a_{0} + a_{1} x + a_{2} \sin x + a_{3} e^{x}
$$
Linear system:
$$
\begin{align} 
  \mathbf{A} \, a &= y \\
\left[
\begin{array}{cccc}
 1 & x_1 & \sin \left(x_1\right) & e^{x_1} \\
 1 & x_2 & \sin \left(x_2\right) & e^{x_2} \\
 1 & x_3 & \sin \left(x_3\right) & e^{x_3} \\
 1 & x_4 & \sin \left(x_4\right) & e^{x_4} \\
 1 & x_5 & \sin \left(x_5\right) & e^{x_5} \\
 1 & x_6 & \sin \left(x_6\right) & e^{x_6} \\
\end{array}
\right]
%
\left[
\begin{array}{cccc}
  a_{0} \\
  a_{1} \\
  a_{2} \\
  a_{3}
\end{array}
\right] 
%
& =
%
\left[
\begin{array}{cccc}
  y_{1} \\
  y_{2} \\
  y_{3} \\
  y_{4} \\
  y_{5} \\
  y_{6} 
\end{array}
\right]
%
\end{align}
$$
Least squares solution:
$$
 a_{LS} = \left\{
a \in \mathbb{C}^{4} \colon
\lVert
 \mathbf{A} \, a - y
\rVert_{2}^{2}
\text{ is minimized}
\right\}
$$
Solution option: normal equations
$$
\begin{align}
%
 \mathbf{A}^{*} \mathbf{A} \, a &= \mathbf{A}^{*} y \\
%
\left[ \begin{array}{llll}
  \sum 1 & \sum x & \sum sin(x) & \sum e^{x} \\
  \sum x & \sum x^2 & \sum x sin(x) & \sum x e^{x} \\
  \sum sin(x) & \sum x sin(x) & \sum sin^{2}(x) & \sum sin(x) e^{x} \\
  \sum e^{x} & \sum e^{x} x & \sum e^{x} sin(x) & \sum e^{2x} \\
\end{array} \right]
\left[
\begin{array}{cccc}
  a_{0} \\
  a_{1} \\
  a_{2} \\
  a_{3}
\end{array}
\right] 
&= 
\left[
\begin{array}{cccc}
  \sum y \\
  \sum x y \\
  \sum y\sin(x) \\
  \sum y \exp(x)
\end{array}
\right] 
%
%
\end{align}
$$
Solution
$$ 
  a_{LS} = \left( \mathbf{A}^{*} \mathbf{A} \right)^{-1} \mathbf{A}^{*}\, y
$$
</s>