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.