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What I want to do in this video is to explore the notion of a matrix

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Outside the context of a surprisingly good movie that involves Keanu Reeves

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and it is actually the first of three, I guess we could call it three movies combined the Matrices

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And in fact there is a relationship between the movie, which is about a virtual reality constructed by super smart computers

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and the relation and the notion of what a matrix is when you study it in mathematics.

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Or when you study it in computer science, the connection really is that matrixes are used a lot when you are

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assimilating things or constructing things in computer science, especially in frankly, computer graphics.

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Then, the super intelligent robots that made the matrix in the movie Matrix probably using matrixes to in order to do it.

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If they actually did exist.

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Now, what is a matrix then?

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Well, that is a fairly simple answer: its just a rectangular array of numbers.

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So, for example, this right over here: 1, 0, -7, pi, 5 and ... 11. This is a matrix.

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This is a matrix where 1, 0, -7, pi ... each of those are entries in the matrix. This matrix right over here has 2 rows and 3 columns.

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And, because it has 2 rows and 3 columns, people will say that it is a 2 x 3 matrix.

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Whenever they say ' this is a something by something matrix ', they're telling you that it has 2 rows.

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So you see the 2 rows right over there and they are telling you

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And they are telling you that it has has 3 columns, you see the 3 columns right there.

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I can give you other examples of matrixes. So I could have a 1 x 1 matrix ...

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So I could have the matrix 1.

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This right over here is a 1 x1 matrix. It has one row and one column

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I can have a matrix like this ... 3, 7 and 17. What is this? Well, this has 1 row, this is the one row we see right here.

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And it has 3 columns, this is a 1 x 3 matrix.

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I could have a matrix, and I think you see where all of this is going,

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Figuring out the dimensions of a matrix is not too difficult.

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I could have a matrix that looks like this, where its 3, 5, 0, 0, -1, -7

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This 7 is negative 7.

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This right over here has 3 rows, so its 3 rows and it has 2 columns.

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So we would call it a 3 by 2 matrix,

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let me do that in the same colour, we would call it a 3 by 2 matrix, 3 rows, and 2 columns.

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So fair enough, you know that a matrix is just a rectangular array of numbers,

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you can say what its dimensions are, you know that each of these numbers,

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that take one of these positions, we just call these entries,

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But what are matrixes good for?

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I still might not be clear what the connection is between this and this right over here.

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And the most fundamental level well these are just ways to, a compact representation,

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of a bunch of numbers, its a way of representing information.

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They become very valuable in computer graphics because these numbers could represent the colour intensity at a certain point

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They could represent whether or not an object is there at a certain point.

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And as we develop an algebra around matrixes, and when we talk about developing an algebra around matrixes,

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We are going to talk about operations that we are going to perform on matrixes.

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That we would normally perform with numbers.

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So we are going to learn how or essentially define how to multiply matrixes, add matrixes,

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How to even take the inverse of a matrix and coming up with an algebra, how we manipulate these things,

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It'll become very useful in the future when you're trying to write a computer graphics program,

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or you're trying to do an economics simulation, or a probability simulation,

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to say oh, I have this matrix that represents where different particles are in space,

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or this matrix represents the state of some type of game.

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And I know the algebra of matrixes, and I know ways of doing it very efficiently,

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So that I can, multiply a bunch of them, or I could come run a simulation,

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and I could actually come up with useful results.

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So that's all matrixes are, but as you'll see through this, we can define operations on them.

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And later on, when you take a linear algebra course in college, you learn a lot more of the depth,

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of how they can be applied, and what you can use them to represent.