Representing systems of any number of equations with matrices
About this video
Building on the 3 by 3 case, the video writes the system 2x + y = 9 and 3x - y = 5 as a 2 by 2 coefficient matrix times the vector (x, y) equal to (9, 5). It then generalises: any system of N equations in N unknowns can be written as Ax = b, with A an N by N matrix and x and b N by 1 vectors, which can be seen as a transformation we'd like to undo.
You'll be able to
- write a system of two equations in two unknowns as a matrix equation.
- state the dimensions of A, x and b in Ax = b for N equations and N unknowns.
- multiply a 2 by 2 matrix by a 2 by 1 vector to check a representation.
- describe solving Ax = b as finding the vector that the transformation A sends to b.
Key words
- coefficient matrix
- The matrix A whose columns hold the coefficients of each variable.
- matrix vector equation
- An equation of the form Ax = b, where A is a matrix and x and b are vectors.
- vector of unknowns
- The vector x that lists the variables, such as (x, y).
- transformation
- Viewing a matrix as an action that moves one vector to another.
Transcript
- [Instructor] 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 Xs, the Ys, and the Zs as its various columns. and then you're going to multiply that times this vector, which is really the vector of the unknown variables, and this is a three-by-one vector. And 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 gonna 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 2x plus y is equal to nine, and we had 3x minus y is equal to five. I encourage you, 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.
So the matrix that represents the coefficients is going to be a two-by-two 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 two-by-one vector, and then that's going to be equal to a vector that represents the constants on the right-hand side, and obviously we have two of those.
So that's going to be a two-by-one 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, 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.
And then we multiply it times the vector of the variables, X, Y, and then last but not least, you have this nine and this five over here, nine and five. And I encourage you and 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. Sum N-by-N matrix A, N by N, times sum N-by-one vector X. This isn't just the variable X. This is a vector X that has N dimensions to it.
So times sum N-by-one vector X is going to be equal to sum N-by-one 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, and 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.
And as I talked about in the previous video, what's interesting about this is you could think about, for example, in 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 as saying, okay, I have some unknown vector in the coordinate plane and I'm transforming it using this matrix to get this vector nine five. And so I have to figure out what vector, when transformed in this way, gets us to nine five, and 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? Which you'll see is actually very useful if you ever become a data scientist, or you go into computer science, or if you go into computer graphics of some kind.
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