Under Over-Determined Systems

University
Massachusetts Institute of Technology
Course
Multivariable Calculus
Pages

1

Academic year

2022
Author

Sporkz
Views

22

p {margin: 0; padding: 0;} .ft00{font-size:20px;font-family:Arial;color:#000000;} .ft01{font-size:18px;font-family:ArialMT;color:#000000;} .ft02{font-size:18px;line-height:22px;font-family:ArialMT;color:#000000;} Under over-determined systems OK, so let's think about what could happen. So I can go back to my picture. And in my firsttwo planes I determined a line, and now I have a third plane. And maybe my third plane isactually parallel to the line but doesn't pass through it. Well then, there aren't anysolutions. In order to solve a system of equations, I need to be in the first two planes. Therefore, Ineed to be in that vertical line. However, the line is red and does not appear as red on thisslide. Also, it needs to be in the third plane; however, the line is parallel to the third planeand 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 theline is contained in the plane. If so, any point on that line will solve automatically the thirdequation. If you try solving a system of equations manually, you will notice that if you substitute thevariables 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 additionalinformation. 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 bedetermined from the first two. For example, if you are able to determine from the first twoequations that x plus z equals 1, then it is impossible for x plus z to also equal 2 in the thirdequation. Another way of saying this is that "this" picture represents an instance in which anumber (x + z) can be found in an equation that is equal to another number (2). This isimpossible. 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, unlessthe third plane is parallel to the line where P1 and P2 intersect, there are two subcases. Ifthe line of intersections of P1 and P2 is contained in P3, in the third plane, then there areinfinitely many solutions. Namely, any point on that line will automatically solve the thirdequation.