![]() "Positioning" this line (or plane) such that it passes through $\bar \x$ is equivalent of centering in the algebraic argument above. ![]() The simplest way to find it is to reduce the matrix to its simplest form. Use this free online algebra calculator to find the rank of a matrix of 3x3 dimension. the dimensionality of the subspace is always $n-1$ this only works because we assume that this line (and plane) can be "moved around" in order to fit our points. In linear algebra, Matrix rank is the maximum number of independent row or column vectors in the matrix. The geometric intuition that I alluded to in the comments above is that one can always fit a 1D line to any two points in 2D and one can always fit a 2D plane to any three points in 3D, i.e. The unbiased estimator of the sample covariance matrix given $n$ data points $\newcommand$.
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