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Current time:0:00Total duration:4:29

What I want to do
in this video is explore the notion
of a matrix outside of the context of a
surprisingly good movie that involves Keanu Reeves. And it's actually
the first of three. I guess we could call the three
movies combined The Matrices. And there is a relationship
between the movie, which is about a virtual
reality constructed by super-smart
computers, and the notion of what a matrix is when
you study it in mathematics, or when you study it
in computer science. And the connection
really is that matrices are used a lot when you
are simulating things or when you're constructing
things in computer science, especially in, frankly,
computer graphics. So the super-intelligent
robots that made the matrix in
the movie Matrix were probably using
matrices in order to do it, if they actually did exist. Now, what is a matrix then? Well, that's a
fairly simple answer. It's just a rectangular
array of numbers. So for example, this
right over here. If I have 1, 0, negative 7,
pi, 5, and-- I don't know-- 11, this is a matrix. This is a matrix where 1, 0,
negative 7, pi-- each of those are an entry in the matrix. This matrix right over
here has two rows. And it has three columns. And because it has two
rows and three columns, people will often say that
this is a 2 by 3 matrix. Whenever they say it's
something by something matrix, they're telling you that it has
two rows-- so you see the two rows right over there. And they are telling you
that it has three columns. You see the three
columns right over there. I could give you other
examples of a matrix. So I could have a 1 by 1 matrix. So I could have the matrix 1. This right over here
is a 1 by 1 matrix. It has one row, one column. I could have a matrix
like this-- 3, 7, and 17. What is this? Well, this has one row. This is the one row
that we see here. And it has three columns. This is a 1 by 3 matrix. I could have a matrix-- and I
think you see where all of this is going. Figuring out the dimensions of
a matrix are not too difficult. I could have a matrix that looks
like this, where it's 3, 5, 0, 0, negative 1, negative 7. This right over
here has three rows. So it's three rows,
and it has two columns. So we would call this a 3 by 2. Let me do that in
that same color. We would call it a 3
by 2 matrix, three rows and two columns. So fair enough. You know that a matrix is just
a rectangular array of numbers. You can say what
its dimensions are. You know that each
of these numbers that take one of these positions--
we just call those entries. But what are matrices good for? I still might not be
clear what the connection is between this and
this right over here. And at the most
fundamental level, this is just a
compact representation of a bunch of numbers. It's a way of
representing information. They become very valuable
in computer graphics because these numbers could
represent the color intensity at a certain point. They could represent
whether an object is there at a certain point. And as we develop an
algebra around matrices, and when we talk about
developing an algebra around matrices, we're going
to talk about operations that we're going to perform on
matrices that we would normally perform with numbers. So we're going to
essentially define how to multiply matrices,
how to add matrices. We'll learn about taking
an inverse of a matrix. And by coming up with an
algebra of how we manipulate these things, it'll
become very useful in the future when you're trying
to write a computer graphics program or you're trying to
do an economic simulation or a probability
simulation, to say, oh, I have this matrix
that represents where different particles
are in space. Or I have this matrix
that represents the state of some
type of a game. And I know the
algebra of matrices. And I know ways of doing
it very efficiently so that I can multiply
a bunch of them. Or I could come
run a simulation, and I can actually come
up with useful results. So that's all matrices are. But as you'll see
through this, we can define operations on them. And then later on, when
you take a linear algebra course in college, you'll
learn a lot more of the depth of how they can be
applied and what you can use them to represent.