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- [Instructor] In a previous video,

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we saw that if you have a system

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of three equations with[br]three unknowns like this,

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you can represent it as[br]a matrix vector equation,

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where this matrix right over here

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is a three-by-three matrix.

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That is essentially a coefficient matrix.

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It has all of the coefficients of the Xs,

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the Ys, and the Zs as its various columns.

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and then you're going to[br]multiply that times this vector,

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which is really the vector[br]of the unknown variables,

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and this is a three-by-one vector.

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And then you would result

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in this other three-by-one[br]vector, which is a vector

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that contains these constant[br]terms right over here.

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What we're gonna do in[br]this video is recognize

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that you can generalize this phenomenon.

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It's not just true with a system

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of three equations with three unknowns.

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It actually generalizes to[br]N equations with N unknowns.

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But just to appreciate that[br]that is indeed the case,

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let us look at a system of two[br]equations with two unknowns.

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So let's say you had 2x[br]plus y is equal to nine,

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and we had 3x minus y is equal to five.

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I encourage you, pause this video

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and think about how that[br]would be represented

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as a matrix vector equation.

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All right, now let's[br]work on this together.

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So this is a system of two[br]equations with two unknowns.

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So the matrix that represents[br]the coefficients is going

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to be a two-by-two matrix

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and then that's going to be multiplied

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by a vector that represents[br]the unknown variables.

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We have two unknown variables over here.

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So this is going to be[br]a two-by-one vector,

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and then that's going[br]to be equal to a vector

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that represents the constants[br]on the right-hand side,

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and obviously we have two of those.

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So that's going to be a[br]two-by-one vector as well.

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And then we can do exactly what we did

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in that previous example[br]in a previous video.

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The coefficients on the[br]X terms, two and three,

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and then we have the[br]coefficients on the Y terms.

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This would be a positive one

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and then this would be a negative one.

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And then we multiply it times the vector

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of the variables, X, Y,

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and then last but not least,

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you have this nine and this[br]five over here, nine and five.

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And I encourage you and multiply this out.

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Multiply this matrix times this vector.

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And when you do that and you[br]still set up this equality,

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you're going to see that[br]it essentially turns

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into this exact same system

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of two equations and two unknowns.

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Now, what's interesting about this is

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that we see a generalizable form.

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In general, you can represent a system

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of N equations and N unknowns in the form.

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Sum N-by-N matrix A,

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N by N,

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times sum N-by-one vector X.

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This isn't just the variable X.

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This is a vector X that[br]has N dimensions to it.

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So times sum N-by-one vector X is going

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to be equal to sum N-by-one vector B.

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These are the letters that[br]people use by convention.

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This is going to be N by one.

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And so you can see in[br]these different scenarios.

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In that first one, this is[br]a three-by-three matrix.

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We could call that A,

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and then we could call this the vector X,

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and then we could call this the vector B.

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Now in that second scenario,

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we could call this the matrix A,

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we could call this the vector X,

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and then we could call this the vector B,

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but we can generalize[br]that to N dimensions.

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And as I talked about[br]in the previous video,

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what's interesting about this[br]is you could think about,

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for example, in this[br]system of two equations

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with two unknowns, as all[br]right, I have a line here,

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I have a line here,

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and X and Y represent the[br]intersection of those lines.

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But when you represent it this way,

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you could also imagine it as saying,

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okay, I have some unknown[br]vector in the coordinate plane

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and I'm transforming it using this matrix

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to get this vector nine five.

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And so I have to figure out what vector,

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when transformed in this[br]way, gets us to nine five,

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and we also thought about it[br]in the three-by-three case.

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What three-dimensional vector,[br]when transformed in this way,

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gets us to this vector right over here?

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And so that hints,

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that foreshadows where[br]we might be able to go.

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If we can unwind this[br]transformation somehow,

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then we can figure out what[br]these unknown vectors are.

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And if we can do it in two[br]dimensions or three dimensions,

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why not be able to do it in N dimensions?

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Which you'll see is actually very useful

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if you ever become a data scientist,

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or you go into computer science,

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or if you go into computer[br]graphics of some kind.