Let's learn about matrices. So, what is a, well, what I do I mean when I say matrices?
Well, matrices is just the plural for matrix.
Which is probably a word you're familiar with more because of Hollywood than because of mathematics.
So, what is a matrix? Well, it's actually a pretty simple idea.
It's just a table of numbers. That's all a matrix is.
So, let me draw a matrix for you.
I don't like that toothpaste colored blue, so, let me use another color.
This is an example of a matrix. If I said, I don't know I'm going to pick some random numbers;
Five, one, two, three, zero, minus five. That is a matrix.
And all it is is a table of numbers and, oftentimes if you want to have a variable for a matrix, you
use a capital letter. So, you could use a capital 'A'.
Sometimes in some books they make it extra bold. So it could be a bold 'A', would be a matrix.
And, just a little bit of notation, So, they would call this matrix. Or, we would call
this matrix, just by convention, you would call this a two by three matrix.
And, sometimes they actually write it '2 by 3' below the bold letter they use to represent the matrix
What is two? And, what is three?
Well, two is the number of rows. We have one row, two row. This is a row, this is a row.
We have three columns; one, two , three.
So, that's why it's called a two by three matrix.
When you say, you know, if I said, if I said that B, I'll put it extra bold.
If B is a five by two matrix, that means that B would have, I can, let me do one
I'll just type in numbers; zero, minus five, ten.
So, it has five rows, it has two columns.
We'll have another column here. So, let's see; minus ten, three,
I'm justing putting in random numbers here. Seven, two, pi.
That is a five by two matrix.
So, I think you'd now have a kind of a convention that all a matrix is is a
table of numbers. You can represent it when you're doing it in variable form
you represent it as bold face capital letter. Sometimes you'd write two by three there.
And, you can actually reference the terms of the matrix.
In this example, the top example, where we have matrix A.
If someone wanted to reference, let's say, this, this element of the matrix.
So, what is that? That is in the second row. It's in row two.
And, it's in column two. Right?
This is column one, this is column two. Row one, row two.
So, it's in the second row, second column.
So, sometimes people will write that A, then they'll write, you know
two comma two is equal to zero.
Or, they might write, sometimes they'll write a lowercase a,
two comma two is equal to zero.
Well, what is A? These are just the same thing.
I'm just doing this to expose you to the notation, because
a lot of this really is just notation.
So, what is a, one comma three?
Well, that means we're in the first row and the third column.
First row; one, two, three. It's this value right here.
So, that equals two.
So, this is just all notation of what a matrix is;
it's a table of numbers, it can be represented this way.
We can represent its different elements that way.
So, you might be asking
"Sal, well, that's nice, a table of numbers with fancy
words and fancy notations. But, what is it good for?"
And that's the interesting point.
A matrix is just a data representation. It's just a way of writing down data.
That's all it is. It's a table of numbers.
But, it can be used to represent a whole set of phenomenon.
And if you're doing this in you Algebra 1 or your Algebra 2 class
you're probably using it to represent linear equations.
But, we will learn, later, that it, and I'll do a whole set of videos
on applying matrices to a whole bunch of different things.
But, it can represent, it's very powerful and if you're doing
computer graphics, that matrixes...The elements can represent pixels on your screen,
they can represent points in coordinate space,
they can represent...Who knows!
There's tonnes of things that they can represent.
But, the important thing to realize is that a matrix
isn't, it's not a natural phenomenon.
It's not like a lot of the mathematical concepts we've been looking at.
It's a way to represent a mathematical concept.
Or, a way of representing values. But you kinda have to
define what it's representing.
But, lets put that on the back burner a little bit
in terms of what it actually represents.
And the, oh, my wife is here. She's looking for our filing cabinet.
But anyway, back to what I was doing.
So, so, lets put on the back burner what a matrix is
actually representing. Let's learn the conventions.
Because, I think, uhm, at least initially, that tends to be
the hardest part, How do you add matrices?
How do you multiple matrices? How do you invert a matrices?
How do you find the determinant of a matrix?
I know all of those words might sound unfamiliar. Unless,
you've already been confused by then in your algebra class.
So. I'm gonna teach you all of those things first.
Which are all really human-defined conventions.
And then, later on, I'll make a whole bunch of videos on the intuition behind them,
and what they actually represent. So, let's get started.
So, lets say I wanted to add these two matrices.
Let's say, the first one, let me switch colors. Let's say,
I'll do relatively small ones, just, not to waste space.
So, you have the matrix; three, negative one, I don't know,
two, zero. I don't know, let's call that A, capital A.
And let's say matrix B, and I'm just making up numbers.
Matrix B is equal to; minus seven, two, three, five.
So, my question to you is: What is A,
so I'm doing it bold like they do in the text books, plus
matrix B? So, I'm adding two matrices. And, once again
this is just human convention. Someone defined how matrices add.
They could've defined it some other way. But, they said;
we're gonna make matrices add the way I'm
about to show you because it's useful for a whole set of phenomenon.
So, when you add two matrices you essentially just add
the corresponding elements. So, how does that work?
Well, you add the element that's in row one column one with
the element that's in row one column one. Alright, so, it's
three plus minus seven. So, three plus minus seven.
That'll be the one-one element. Then, the row one column two element
will be minus one plus two.
Put parenthesis around them so you know that these are
separate elements. And, you could guess how this keeps going.
This element will be two plus three. This element, this last element will be zero plus five.
So, that equals what? Three plus minus seven, that is minus four.
Minus one plus two, that's one. Two plus three is five. And,
zero plus five is five. So, there we have it, that is how we humans have defined the addition of two matrices.
And, by this definition, you can imagine that this is going to be the same thing
as B plus A. Right? And remember, this is something we have to think about
because we're not adding numbers anymore. You know one plus two is the same as
two plus one. Or, any two normal numbers, it doesn't matter what order you
add them in. But matrices it's not completely obvious. But, when you define it in this way
it doesn't matter if we do A plus B or B plus A. Right?
If we did B plus A, this would just say negative seven plus three.
This would just say two plus negative one. But, it would come out to the same values.
That is matrix addition.
And, you can imagine, matrix subtraction, it's essentially the same thing.
We would...Well, actually let me show you. What would be A minus B?
Well, you can also view that, this is capital B, it's a matrix
that's why I'm making it extra bold. But, that's the same thing as;
A plus minus one, times B. What's B? Well, B is;
minus seven, two, three, five. And, when you multiply
a scalar, when you just multiply a number times the matrix,
you just multiply that number times every one of its elements.
So, that equals A, matrix A, plus the matrix, we just multiply
the negative one times every element in here. So, seven,
minus two, minus three, five. And then we can do
what we just did up there. We know what A is. So,
this would equal, let's see, A is up here. So, three plus
seven is ten, negative one, plus negative two is minus three,
two plus minus three is minus one and zero plus five is five.
And, you didn't have to go through this exercise right here.
You could have, literally, just subtracted these elements from these elements
and you would have gotten the same value.
I did this because I wanted to show you also that multiplying
a scalar times, or just a value or a number, times a matrix
is just multiplying that number times all of the elements of that matrix.
And, so what...By this definition of matrix addition what do we know?
Well, we know that both matrices have to be the same size,
by this definition of the way we're adding. So, for example
you could add these two matrices, You could add, I don't know,
one, two, three, four, five, six, seven, eight, nine to this matrix;
to, I don't know, minus ten, minus one hundred, minus one thousand.
I'm making up numbers. One, zero, zero, one ,zero, one.
You can add these two matrices. Right?
Because they have the same number of rows and the same number of columns.
So, for example, if you were to add them. The first term up here would be one plus minus ten,
so, it would be minus nine. Two plus minus one hundred, minus ninety-eight.
I think you get the point. You'd have exactly nine elements and you'd have three rows of three columns.
But, you could not add these two matrices. You could not add...
Let me do it in a different color, just to show it is different,
You could not add, this blue, you could not add this matrix;
minus three, two to the matrix; I don't know, nine, seven.
And why can you not add them?
Well, they don't have corresponding elements to add up.
This is a one row by two column, this is one by two
and this is two by one. So, they don't have the same dimensions
so we can't add or subtract these matrices.
And, just as a side note, when a matrix has...when one of its
dimensions is one. So, for example, here you have one row
and multiple columns. This is actually called a row vector.
A vector is essentially a one dimensional matrix, where one
of the dimensions is one. So, this is a row vector and similarly,
this is a column vector. That's just a little extra terminology
that you should know. Uhm, if you take linear algebra and calculus
your professor might use those terms and it's good to be
familiar with it. Anyway, I'm pushing eleven minutes, so I will continue this in the next video. See you soon.