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The assumption of linear least squares is that there is a linear relationship between our measurements  and the variables to be estimated 
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(1) |
For this example let us assume that our measurements are given in Table 1 and you can see them plotted in Figure 1.
| x |
-3.0 |
-2.5 |
-2.0 |
-1.5 |
-1.0 |
-0.5 |
0.0 |
0.5 |
1.0 |
1.5 |
| z |
-1.0 |
-0.25 |
0.0 |
0.25 |
0.4 |
0.7 |
1.0 |
1.1 |
1.4 |
1.8 |
Table 1: Example Data
The linear least squares solution to fit the given data is given by the equation
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(2) |
The only not so obvious step before using a tool like Matlab, is to form the  matrix, which is a combination of an identity vector and  as column vectors, such that
This is clarified by looking at the example code in Matlab, LinearLeastSquares.m. A plot of fitting the measurement data with a line such that it minimizes the the mean square of the error is shown in Figure 1.
The equation of the line to fit this data is then
Figure 1: Linear Fit of Example Data (Matlab)
Figure 1: Linear Fit of Example Data (rlplot)
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![$\displaystyle A = [ \,\, 1 \,\, \vert \,\,x \,\,] $](http://images.physicslibrary.org/cache/objects/245/l2h/img7.png "$\displaystyle A = [ \,\, 1 \,\, \vert \,\,x \,\,] $")
![\includegraphics[scale=.6]{LinearLeastSquares2.eps}](http://images.physicslibrary.org/cache/objects/245/l2h/img9.png "\includegraphics[scale=.6]{LinearLeastSquares2.eps}")
![\includegraphics[scale=.8]{least_squares.eps}](http://images.physicslibrary.org/cache/objects/245/l2h/img10.png "\includegraphics[scale=.8]{least_squares.eps}")