xb xb span(xfigure xb is the orthogonal projection of onto the linear space spanned by the columns of the matrix according to theorem the projection matrix is given by xpseudo-inverse normal equations where the matrix xin ( is the pseudo-inverse of if happens to be of full column rank (so that none of the columns can be expressed as linear combination of the other columns)then (xx)- xin any casefrom xb py and px xwe can see that satisfies the normal equationsxxb xpy (px) xy ( this is set of linear equationswhich can be solved very fast and whose solution can be written explicitly asb xy ( figure shows the trained learners for various values of ph ht (ugt (xx> data points true underfit correct overfit ( ( figure training data with fitted curves for and the true cubic polynomial curve for is also plotted (dashed line