Vectors
Introduction to Vectors Introduction to Vectors
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- 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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