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Linear Algebra for Machine Learning and Data Science
34 hours 5 mins
Now that you learned how systems of linear equations are represented by lines in the plane, and that non-singularity equates to these lines intersecting at a unique point, I'm going to show you an even simpler way to visualize singularity and non-singularity. It involves slightly simplifying the system. Recall that these are the three systems you've been studying, where system 1 has a unique solution, so it's complete and non-singular, system 2 has infinitely many solutions, so it's redundant and singular, and system 3 has no solutions, so it's contradictory and singular. Now forget for a moment about the system being complete, redundant, or contradictory, and let's focus on the singularity and non-singularity. And the reason for this is that these are the terms we're going to be studying during the whole class. Note that system 2 and 3, the singular ones, are very similar, since both consist of parallel lines. Difference lies that in system 2 these lines are the same, and in system 3 they're spaced apart, but both are fundamentally different from the non-singular system 1, since in system 1 the two lines are not parallel. So can you somehow contract systems 2 and 3 into one, in order to really separate singular and non-singular systems into two buckets? And the answer is yes. And the way to do this is to look at the constants of the systems of equations. These constants are the numbers in the equations that are not accompanying the variables a or b. For instance, in system 1 they are 10 and 12, in system 2 they're 10 and 20, and in system 3 they're the 10 and the 24. So let's turn all these constants for all these three systems into 0, and see what happens to the plots. So, when you turn these constants to 0, the plots now become this. Why? Because the new systems always have the point 0,0 as a solution, so they must pass by the origin. Notice that if I were to set a and b to be 0 in any of these equations, the equations would work, so 0,0 is a solution to all of them. Now notice that system 1 is still a pair of intersecting lines, so it still has a unique solution, so it's still complete and non-singular. System 2 is still a pair of identical lines, so it still has infinite solutions, so it's redundant and singular. But notice what happened to system 3. It went from a pair of non-intersecting parallel lines, to a pair of identical lines, so now it has infinitely many solutions instead of none. So it went from contradictory to redundant. However, it stayed singular as before, and that's what matters. So, in conclusion, the constants in the system don't matter when it comes to determining if the system is singular or non-singular. Now, for the rest of this course, singularity and non-singularity will be the important concepts, and completeness, redundancy, and contradiction will be used much less. So, for this reason, you can now start considering systems of equations where the constants are always zero, and these are much simpler. Furthermore, the geometric interpretation of these systems will be pairs of lines that go through the origin.