In this first, what I'll call this is my first real linear
algebra video, I want to introduce you to some concepts
that you're going to see over and over in linear algebra,
and if they're not introduced to you properly in the
beginning, they can be very confusing.
But hopefully over the course of this video, you'll
understand that they're pretty straightforward ideas.
If anything, in linear algebra, the field, or the
gods of linear algebra, are kind of experts in to some
degree stating simple and obvious things in convoluted
and Byzantine ways.
But it does that for a reason; it gives it
mathematical rigor.
So the first thing I want to introduce you to is the set
notation for real numbers and then multiple dimensions of
real numbers.
So if you ever see this, an R with an extra backbone like
that, or sometimes it's written as just a really
bold-faced R if it's maybe typed in a book, a really
bold-faced R, this just means, the set of all real numbers.
And I probably should make a few videos where I classify
numbers by reals and complex and irrational and rational
and whatever else, and there's actually very formal ways of
defining the set of all real numbers, but the simple thing
for me is all numbers except complex ones.
So this is the same thing as
everything but complex numbers.
So pi is a real number, e is a real number, square root of 2
is a real number, 3 is a real number, minus 3 is a real
number, minus pi is a real number, But 1 plus 2i is not a
real number.
It's not in the set of reals; that's a complex number.
So everything but complex.
So pretty much all regular numbers that you ever dealt
with before you learned what i was.
That's the set of real numbers.
Now, the next thing you'll see when you start discussing
linear algebra is someone will write-- they won't just write
the set of all real numbers, they'll write Rn.
And you're like, gee, what is that?
Are they somehow taking the exponent of
all of the real numbers?
And actually on some level, that is what they're doing,
but what they're saying here is they're taking the set of
all ordered sets of real numbers.
So what do I mean by that?
So let me do an example where I don't just have Rn,
So R2 is the set of all lists of real numbers, and let me
make one of the lists.
So let's say this has one number and then
it has another number.
So it's the set of -- let me write it this way, so I can
write in the set of notations.
It's the set of all-- we could call these tuples.
Sometimes it's called a 2-tuple, which just means a
list of two numbers.
That's all it means.
It just means a list of two numbers.
So R2 is a list of all ordered 2-tuples, so all ordered lists
of two numbers.
Let me actually write down the word ordered because that's
important, so ordered list of 2-tuples.
So, for example, this would be-- well, let me
write it this way.
This would be a different-- when I say ordered, I'm saying
that this is fundamentally different than that, that
these aren't the same 2-tuple.
Each of these is a 2-tuple, but R2 is a set of all
2-tuples such that-- that line there just means such that or
maybe for which-- each of the numbers-- and I'll
write it this way.
I could just write-- well, I could write it a couple of
ways, but I'll write it in the harder to understand way, just
so you get used to it.
Such that xi is a member of the reals.
So member of, that's what this little character means.
So that each xi is a member of the reals.
So what do you you mean xi?
Well, xi just means any of these x's, and I'll write for
i is less than or equal to 2 and greater than or equal to
1, and i is an integer.
Now, I could have written this a bunch of different ways.
This is almost silly how much effort I'm taking to write it
this way, but I could have written it like this, too.
I could have written that R2 is equal to the set of all
ordered tuples, x1 x2 such that-- I could have just
written such that x1 and x2 are a member of the reals.
I could have written it that way.
That would have been an easier way to write it.
But either way, I think you get the idea.
It's all of the combinations of two things.
So I said in the beginning, this kind of is a squared
operator. and why do they write it as
a superscript there?
And my gut sense is if you think about it, you have an
infinite number of real numbers.
I won't go into things like accountability and things like
that, but you have a very large number of real numbers.
But let's say that you only had n real numbers, right?
Let's say that we weren't dealing with reals.
Just for sake of argument, let's say that you had like
100 real numbers, right?
So that means that there would've been 100
possibilities there, and then there's another 100
possibilities there.
So now your set of all the possibilities of 2-tuples,
well, you'd have 100 times 100 possibilities, so you'd almost
have the squared number of possibilities.
So however many possibilities you have for R, you have that
many squared possibilities for R2, which doesn't make a lot
of sense, because you have an infinite number of
possibilities, but you have that many more possibilities.
You have an infinite number of possibilities here and an
infinite number of possibilities here.
So it's almost a greater infinity of potential
possibilities.
Now, what's R3?
R3 is just the same thing.
R3 means you're going to have three points or three numbers.
I want to be careful not to just say points.
We can graph these things, and, for example, R2, point in
R2 can normally be graphed on graph paper just in Cartesian
coordinates.
Points in R3 can be graphed in three dimensions.
Points in just R, or we could just say R1, you could just
graph those on a number line.
I can have a number line like that, and if I had to graph
pi, you know, that's 0.
I just say, hey, right there, that's pi.
And if i had to graph e, I'd say, hey, right
there, that's e.
And if I had to graph 1, I'd say, hey, that's 1.
So you could graph them, but I want to be careful.
You don't have to graph them.
They don't have to correspond to actual
physical points on a graph.
But anyway, when you talk about R3, you're literally
just talking about all the possibilities where you have
an ordered set of three numbers, where all three
numbers are a member of the real numbers.
That's all you're saying.
Now, I want to introduce you to another
definition of a vector.
Now, a vector, you've seen before.
You've seen it in physics, where you said, oh, it's
something that has a magnitude and a direction.
You saw it in calculus, and we plotted them and whatever not.
But I'm going to be very formal and very abstract and
very broad with the vector right now because the beauty
of linear algebra is that it doesn't just apply to things
like physics, and it doesn't just apply-- well, it does
apply a lot to physics, but it doesn't just apply to that.
In linear algebra, you can apply it to anything.
It doesn't have to apply to things that just graphs in
three dimensions.
And we'll talk more about that in the future, but that's why
I'm trying to stay pretty abstract.
So I'm going to define a vector in Rn, a
vector to be in Rn.
Actually, let me be careful.
I haven't even defined what Rn is.
I just defined R2 and R3, so let me define Rn.
Rn is equal to the set of all n-tuples, ordered n-tuples.
So you have x1, x2, all the way over to xn.
So you're going to have n things; so if this was 100,
you'd have 100 things here.
So this is an ordered n-tuple.
This is an ordered n-tuple, such that for each xi, where
xi is one of these, it is a member of the real, so each of
these has to be a real number.
And then I'll just say for i is less than or equal to n,
greater than or equal to 1.
So all that's saying is that x1 is a member of the reals,
x2 is a member of the reals, all the way up to xn is a
member of the reals.
Rn is the set of all of these possible ordered n-tuples or
ordered sets of n numbers.
So what is a vector?
What is a vector in Rn?
Well, a vector in Rn really is just a particular instance of
one of these n-tuples.
I'll just call it an ordered list of n real numbers.
And you can represent a vector in a bunch of different ways.
You could represent it like this.
You could represent it like-- let me do it a bunch of
different ways.
A two-dimensional vector, you could represent like x1, x2,
kind of like a coordinate.
You could maybe represent it with brackets like that.
This is just syntax.
This is just different ways of representing the same
information, the same idea.
And for most of what we talk about in linear algebra, we're
going to represent them as essentially these columns, and
in the very near future, we'll represent
them as rows as well.
But I'm going to define my vector.
A vector as an Rn is an ordered list of n real
numbers, and I define-- and this is an example.
So it would be v1, v2, all the way to vn, where each of these
is a member of the real numbers.
And just to make sure you understand the notation or the
terminology, each of these is called a
component of our vector.
Actually, I want to be very careful here, because when we
write vectors, you want to make sure you
have a nice bold v.
I won't make all my v's in the future that bold, but that's
how, when you're reading a math book, they differentiate
between just a vector and just a regular quantity , right?
These regular quantities are just written non-bolded, while
the vectors are bolded.
Now, I'm going to define two operations for vectors in Rn.
I'm going to define addition, and these are definitions.
I could've defined them in any arbitrary way, but these I
think you'll find to be somewhat natural, so these are
definitions.
These are just human abstract constructs that end up
becoming very useful.
So they said, hey, let's make some definitions.
Let's define addition.
So, if I have some vector-- let me write it as a lowercase
a, but it's a really bolded lowercase a, and then that is
equal to a1, a2.
That's all of it's components all the way to an, and I want
to add that.
And then I have vector b-- and I'm going to make that a
really bolded b right there-- is equal to b1, b2, all the
way to bn, I'm going to define addition of these two vectors
to be just the sum of each of their components.
Bolded a, plus bolded b is equal to a new vector, where
you just add each of their components.
It's going to be a1 plus b1 is going to
create this new component.
a2 plus b2, all the way down to an plus bn.
And if you found this somewhat confusing, just think, I'm
almost going through pains to write something very simple in
very confusing terms. If I have two vectors in R2, let's
say I have the vector 2, minus 1, and I want to add that to
the vector 3, 2, I literally just add each of the
corresponding components.
So I add the first components to each other.
So 2 plus 3 is equal to 5.
Minus 1 plus 2 is equal to 1.
That's all my definition of vector addition is for my
vectors in real numbers.
And now the next thing I'm going to define is the idea of
a scalar multiple.
You might remember from physics class, a scalar is
just a regular number, or if your multiplying something by
a scalar, you're just scaling it up.
Maybe I shouldn't use the same word in the definition, but
I'm going to define it as-- let me say c is a scalar.
So notice, I didn't bold the c.
So c times my vector a-- and here I'm going to take pains
to bold it up-- is going to be equal to all of my components
of vector a multiplied by each, multiplied by c.
So a1, a2, all the way to an, and they're each going to be
multiplied by c.
So that's all I define-- every term is going to be
multiplied by c.
And just to give you an example like that, if I wrote
minus 3 times some vector-- I'll just
draw the vector here.
Let me draw a vector in R3.
So let's say the first term is pi, the second term is minus
2, and then the third term is 0, then this is just going to
be equal to a new vector in R3 or you can kind of view it as
a three-dimensional vector, minus 3pi, minus 3 times minus
2 is positive 6, and then minus 3 times 0 is just 0.
So it's that straightforward.
They made this fancy definition, but it's fairly
easy to deal with, I think.
And I'll just make one last definition, and I'll call that
the zero vector.
And depending on what dimension we're in, it just
means a vector with all zeroes.
So the zero vector in Rn, if it's arbitrary, is just a
vector where everything is zero.
And if this is Rn, you just have n components.
So those are all my definitions.
I think I've given you enough to think about right now in
this video.
In the next video, I'm going to show a bunch more of
examples, and I'll actually draw representations of these
vectors, although you don't have to, and we'll kind of
have a little more visual understanding
of what they mean.