In the theorem we are about to prove, the conclusion is that two systems are equivalent. By Definition ESYS this translates to requiring that solution sets be equal for the two systems. So we are being asked to show that two sets are equal . How do we do this? Well, there is a very standard technique, and we will use it repeatedly through the course. If you have not done so already, head to Section SET and familiarize yourself with sets, their operations, and especially the notion of set equality, Definition SE , and the nearby discussion about its use.

One thing to note about outliers is that although we have limited our discussion here to abnormal values in the dependent variable, unusual values in the features of a point can also cause severe problems for some regression methods, especially linear ones such as least squares. The trouble is that if a point lies very far from the other points in feature space, then a linear model (which by nature attributes a constant amount of change in the dependent variable for each movement of one unit in any direction) may need to be very flat (have constant coefficients close to zero) in order to avoid overshooting the far away point by an enormous amount. Hence, points that are outliers in the independent variables can have a dramatic effect on the final solution, at the expense of achieving a lot of accuracy for most of the other points.