Representing systems of any number of equations with matrices | Precalculus | Khan Academy

In a previous video, we saw that if you have a system of three equations with three unknowns, like this, you can represent it as a matrix vector equation. Where this matrix, right over here, is a three by three matrix that is essentially a coefficient matrix. It has all of the coefficients of the x's, the y's, and the z's as its various columns.

Then, you're going to multiply that times this vector, which is really the vector of the unknown variables, and this is a 3 by 1 vector. Then, you would result in this other three by one vector, which is a vector that contains these constant terms right over here.

What we're going to do in this video is recognize that you can generalize this phenomenon. It's not just true with a system of three equations with three unknowns; it actually generalizes to n equations with n unknowns. But just to appreciate that that is indeed the case, let us look at a system of two equations with two unknowns.

So, let's say you had two x plus y is equal to nine, and we had three x minus y is equal to five. I encourage you to pause this video and think about how that would be represented as a matrix vector equation.

All right, now let's work on this together. So, this is a system of two equations with two unknowns. The matrix that represents the coefficients is going to be a 2 by 2 matrix, and then that's going to be multiplied by a vector that represents the unknown variables. We have two unknown variables over here, so this is going to be a 2 by 1 vector.

And then that's going to be equal to a vector that represents the constants on the right-hand side. Obviously, we have two of those, so that's going to be a 2 by 1 vector as well. And then we can do exactly what we did in that previous example.

In a previous video, the coefficients on the x terms are two and three, and then we have the coefficients on the y terms. This would be a positive one, and then this would be a negative one. Then, we multiply it times the vector of the variables x and y, and then last but not least, you have this 9 and this 5 over here—9 and 5.

I encourage you to multiply this out. Multiply this matrix times this vector, and when you do that, and you still set up this equality, you're going to see that it essentially turns into this exact same system of two equations and two unknowns.

Now, what's interesting about this is that we see a generalizable form. In general, you can represent a system of n equations and n unknowns in the form: some n by n matrix A times some n by 1 vector x. This isn't just the variable x; this is a vector x that has n dimensions to it.

So, times some n by 1 vector x is going to be equal to some n by 1 vector b. These are the letters that people use by convention. This is going to be n by one, and so you can see in these different scenarios. In that first one, this is a three by three matrix; we could call that A. Then, we could call this the vector x, and then we could call this the vector b.

Now, in that second scenario, we could call this the matrix A, we could call this the vector x, and then we could call this the vector b. But we can generalize that to n dimensions. As I talked about in the previous video, what's interesting about this is you could think about, for example, this system of two equations with two unknowns as, "All right, I have a line here, I have a line here, and x and y represent the intersection of those lines."

But when you represent it this way, you could also imagine it is saying, "Okay, I have some unknown vector in the coordinate plane, and I'm transforming it using this matrix to get this vector 9, 5." So, I have to figure out what vector, when transformed in this way, gets us to 9, 5.

We also thought about it in the three by three case. What three-dimensional vector, when transformed in this way, gets us to this vector right over here? And so, that hints, that foreshadows where we might be able to go. If we can unwind this transformation somehow, then we can figure out what these unknown vectors are.

And if we can do it in two dimensions or three dimensions, why not be able to do it in n dimensions? What you'll see is actually very useful if you ever become a data scientist or if you're going to computer science or if you go into computer graphics of some kind.