Under over-determined systems
OK, so let’s think about what could happen. So I can go back to my picture. And in my first two planes I determined a line, and now I have a third plane. And maybe my third plane is actually parallel to the line but doesn’t pass through it. Well then, there aren’t any solutions. In order to solve a system of equations, I need to be in the first two planes. Therefore, I need to be in that vertical line. However, the line is red and does not appear as red on this slide. Also, it needs to be in the third plane; however, the line is parallel to the third plane and there is no point at which these lines intersect. Thus, there is no possible way to solve all three equations. However, it is possible that the line is contained in the plane. If so, any point on that line will solve automatically the third equation. If you try solving a system of equations manually, you will notice that if you substitute the variables in one equation into another, eliminate variables, and perform other operations, the third equation will always be the same as the first two. You have gained no additional information. It is as if you had only two equations instead of three. A case of this type would be when the third equation contradicts something that can be determined from the first two. For example, if you are able to determine from the first two equations that x plus z equals 1, then it is impossible for x plus z to also equal 2 in the third equation. Another way of saying this is that “this” picture represents an instance in which a number (x + z) can be found in an equation that is equal to another number (2). This is impossible. In this case, 0 = 0, which is certainly true but not particularly useful. So, you cannot actually solve this system of equations. Let me write that down. So, unless the third plane is parallel to the line where P1 and P2 intersect, there are two subcases. If the line of intersections of P1 and P2 is contained in P3, in the third plane, then there are infinitely many solutions. Namely, any point on that line will automatically solve the third equation. Unless …. the 3^rd plane is parallel to the line where P_1 and P_2, intersect, if line P_1 P_2, is contained in P_3 : infinitely many solutions! (any point on the line is a solution)
