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x and y value for the i-th data sample. Then the model can be written as a system of linear equations:

:

which when using pure matrix notation is written as

:

The vector of estimated polynomial regression coefficients (using ordinary least squares estimation) is

:

assuming m < n which is required for the matrix to be invertible; then since is a Vandermonde matrix, the invertibility condition is guaranteed to hold if all the values are distinct. This is the unique least-squares solution. In other words, it realizes the minimum of the distance between the sample and the corresponding value of the polynomial, , that is, it realizes the minimum
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