00:03
Hello everyone.
00:04
So today we're going to work on a problem and it's going to be, you know, solving another proof and dealing with inverteable matrices and what that means and what it means for a homogenous system of linear equations to have a trivial solution.
00:22
You know, and how can we prove that, given that one homogenous system has only a trivial solution that a similar homogenous system of linear equations also has.
00:32
The trivial solution.
00:33
And as you know, the trivial solution is that the only solution to the system is x equals zero.
00:43
And there's many ways to prove proofs like this, right? or theorems like this.
00:48
But one really common way is to do proof by contradiction.
00:52
And in this situation, i'm going to go about and do this such that i'm going to show the contradiction in both the forward and a backwards direction.
01:00
So just repeat the question so you know it's that let a x equals zero be a homogenous system of n linear equations and n unknowns and let q be an invertible m by n matrix.
01:11
So what they want us to do is prove that a x equals zero has only the trivial solution if and only if q a x equals zero has only the trivial solution itself.
01:22
So we got to show basically that if and only if so let me write this down.
01:29
Show.
01:30
This is what we're going to need to show, right? if and only if a qax equals zero has only the trivial solution, then a x equals zero has only the trivial solution as only the trivial solution as well.
02:10
I'm not write down our solution, right? this is going to be our solution.
02:17
I'm going to write in the top here.
02:19
Proof by contradiction, just so you can remember what is the method we're using to attack this problem.
02:28
Yeah, this is a really popular one too.
02:31
So let's start with going about the forward directions.
02:35
We're going to start by saying, so i'm going to go ahead, just draw an arrow so you can know how we're going to do that.
02:43
Attack this.
02:45
So we're going to let a be a square matrix.
02:48
So let a square matrix, right? and that q is going to also, q is going to be an invertible n by n matrix.
03:08
So let q be an invertible n by n.
03:19
So given that that we are given that a x equals zero has only the trivial solution right so we're going to say that and then we're going to suppose that q a x equal zero for the suppose that the solution for q a x equal zero is not the trivial solution so that's where the contradiction is going to come from because we're going to assume the opposite actually so going by this method is by assuming the opposite of what we're actually trying to prove and by showing that the opposite does have a contradiction in it i'm going to write that down now, we are given a x equals zero, oops, has only trivial solution.
04:12
Suppose that qax equals zero, qax equals zero for only non -trivial solutions.
04:24
We're gonna be like that.
04:25
All right, just to indicate this is a different x, right? therefore, you know, therefore we're gonna be able to say that, 0 equals qax equals qax and we're just reordering oops we're just reordering how we're going to be writing this out right and this is very simple and we all know how to do this moving around the variables but we know that since q is invertible then we can go ahead and left multiply everything in this equation but q inverse, right? so we're going to do that to this right here...