This post explains how to make linear approximation with a simple example. Let’s assume we want to approximate a point cloud with a linear function : 
.
.
The point cloud is given by 
points with coordinates 
. The aim is to estimate 
and 
where 
will fit the point cloud as mush as possible. We want to minimize for each point 
the difference between 
and 
, ie. we want to minimize 
. The matrix form of the system is given by:
points with coordinates . The aim is to estimate and where will fit the point cloud as mush as possible. We want to minimize for each point the difference between and , ie. we want to minimize . The matrix form of the system is given by:![\left[ \begin{matrix} x_1 & 1 \\x_2 & 1 \\... & ... \\x_n & 1 \\\end{matrix} \right].\left[ \begin{matrix} \hat{a} \\\hat{b}\end{matrix} \right]=\left[ \begin{matrix} y_1 \\y_2 \\... \\y_n \\\end{matrix} \right]](https://l.wordpress.com/latex.php?latex=%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20x_1%20%26%201%20%5C%5Cx_2%20%26%201%20%5C%5C...%20%26%20...%20%5C%5Cx_n%20%26%201%20%5C%5C%5Cend%7Bmatrix%7D%20%5Cright%5D.%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%5Chat%7Ba%7D%20%5C%5C%5Chat%7Bb%7D%5Cend%7Bmatrix%7D%20%5Cright%5D%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20y_1%20%5C%5Cy_2%20%5C%5C...%20%5C%5Cy_n%20%5C%5C%5Cend%7Bmatrix%7D%20%5Cright%5D&bg=FFFFFF&fg=000000&s=1 "\left[ \begin{matrix} x_1 & 1 \\x_2 & 1 \\... & ... \\x_n & 1 \\\end{matrix} \right].\left[ \begin{matrix} \hat{a} \\\hat{b}\end{matrix} \right]=\left[ \begin{matrix} y_1 \\y_2 \\... \\y_n \\\end{matrix} \right]")
Let’s define 
![\begin{matrix}A=\left[ \begin{matrix} x_1 & 1 \\ x_2 & 1 \\ ... & ... \\ x_n & 1 \\ \end{matrix} \right] &B=\left[ \begin{matrix} y_1 \\y_2 \\... \\y_n \\\end{matrix} \right] &\hat{x}=\left[ \begin{matrix} \hat{a} \\\hat{b}\end{matrix} \right] \end{matrix}](https://l.wordpress.com/latex.php?latex=%20%5Cbegin%7Bmatrix%7DA%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20x_1%20%26%201%20%5C%5C%20x_2%20%26%201%20%5C%5C%20...%20%26%20...%20%5C%5C%20x_n%20%26%201%20%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%26B%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20y_1%20%5C%5Cy_2%20%5C%5C...%20%20%5C%5Cy_n%20%5C%5C%5Cend%7Bmatrix%7D%20%5Cright%5D%20%26%5Chat%7Bx%7D%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%5Chat%7Ba%7D%20%5C%5C%5Chat%7Bb%7D%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cend%7Bmatrix%7D&bg=FFFFFF&fg=000000&s=1 "\begin{matrix}A=\left[ \begin{matrix} x_1 & 1 \\ x_2 & 1 \\ ... & ... \\ x_n & 1 \\ \end{matrix} \right] &B=\left[ \begin{matrix} y_1 \\y_2 \\... \\y_n \\\end{matrix} \right] &\hat{x}=\left[ \begin{matrix} \hat{a} \\\hat{b}\end{matrix} \right] \end{matrix}")
The system is now given by:
The optimal solution is given by:
Where 
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