Intersections of planes
A linear system is a mathematical model that can be described by an equation in which the variables are unknown but are assumed to depend on one or more independent variables X. A linear system can be classified as either homogeneous or inhomogeneous. Inhomogeneous systems have additional constraints, such as algebraic equations and inequalities, which must be satisfied by the solution of the system. Homogeneous systems do not have any such restrictions. So let’s say we have a 3 3 by 3 linear system. Just to take an example, it doesn’t really matter what we have, but let’s say it is x plus Z equals 1, x plus equals 2, x plus 2y plus 3z equals 3. Well. what does this mean? How do we solve it? It means we want to find x, y and z which satisfies all these conditions. So let’s take a look at the first equation first. The first equation says that our point should be on the plane whose equation is given. The second equation determines that our point should also be on another plane, since the two planes intersect. So in fact, these two equations give us two planes, since we know that the intersections of two planes are lines. Now, well, what happens with the third equation? That’s actually going to be a third plane. In order to solve for the first two equations, we need to be on this line. If we want to solve for a third equation, we also need to be on another plane. In general, three planes intersect in a point. This can be visualized by drawing lines through the first two planes of intersection and then projecting them onto the third plane to find where they meet. And that point is the solution to the linear system. So the line, let’s say, formed by the intersection between the first two planes intersects the third plane in a point, which is going to be the solution. That was a mathematical notation for the intersection between the first two planes. So, one way to solve this problem is to find the solution by drawing pictures and trying to identify where the solution is. But we tend not to use this method in practice if we have the equations available. The solution is provided– so let us use matrix notation. Remember, we saw before that the solution to AX = B is given by X = A inverse B. To get from here to here, multiply on the left by A inverse. A inverse × AX simplifies to X = A inverse B. And once again, it’s A inverse B and not BA inverse. If you try to set up the multiplication, BA inverse does not work. The sizes are not compatible. You cannot multiply across or down.
