Main content

## Linear algebra

### Unit 1: Lesson 1

Vectors- Vector intro for linear algebra
- Real coordinate spaces
- Adding vectors algebraically & graphically
- Multiplying a vector by a scalar
- Vector examples
- Scalar multiplication
- Unit vectors intro
- Unit vectors
- Add vectors
- Add vectors: magnitude & direction to component
- Parametric representations of lines

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# Real coordinate spaces

Created by Sal Khan.

## Want to join the conversation?

- This may sound weird but can there be a negative dimension? Like R^-3?(84 votes)
- The smallest number of dimensions is zero. For example: 3 dimensions is
`{x,y,z}`

. 2 dimensions is`{x,y}`

. 1 dimension is`{x}`

. Zero dimensions is just`{}`

. It is null, there is no information there, and that's as low as you can get.

You can think of dimensionality as like the distance from something. It is an absolute value. You can't be "negative 3 miles" from home, because you'd just be 3 miles from home. Similarly, you can't have "negative 3" dimensions.(29 votes)

- At1:28, Sal talks about tuples. This makes me think of "quintuples," except the "tu" is pronounced differently. Is that where "tuples" comes from? Where does the term "tuples" come from?(14 votes)
- A tuple is a collection of objects but unlike sets tuples are always OEDERED collections of objects.

I do not know if the word originated from quintuple(does it mean ordered collection of 5 things ? )(2 votes)

- In the case of a complex numbers in a tuple, could you define a complex coordinate space? Like C^n?(10 votes)
- Yes, indeed. Such spaces are very useful in mathematical physics as well as interesting from a pure mathematics point of view.(14 votes)

- what is the dimension of [0,0]..?(6 votes)
- R2, there are two components to your vector, which are both without imaginary components.(8 votes)

- So let me get this straight,

if you had a vector with coordinates (5, I (<That is not a one, it is the letter i.)), it could not be defined by R^2 because I is an imaginary number whose square is -1.(4 votes)- You are correct, any vector whose elements are complex numbers is outside of
`ℜⁿ`

, those vectors would belong to the set of complex numbers`ℂⁿ`

; in your case, since your vector has 2 components, it would belong to`ℂ²`

.(12 votes)

- how would the vector v=[0,0] be a vector if it has no direction it is not a line(4 votes)
- We declare <0,0> to be a vector because we want the sum of any two vectors to be a vector. <1, 1> and <-1, -1> are vectors, so we want their sum <0,0> to be a vector as well.(6 votes)

- Sal's pronunciation of "tuples" is really throwing me off. When working in python, I would always pronounce them "tuh-pulls". Is this referring to the same sort of thing as it is in computer programming, or is it something else entirely? Have I been pronouncing it wrong all this time? haha(4 votes)
- Either pronunciation is correct.

Source: https://english.stackexchange.com/questions/12980/how-to-pronounce-tuple(4 votes)

- If i am going to write a vector say in the 10 dimension, i just need to add numbers to the list, 10 numbers vertically?(3 votes)
- Yes, we think of 10-dimension as a list of 10 numbers. The first number tells you the coordinate of the first dimension and so on.(5 votes)

- what about R^0 ?(2 votes)
- That's just the trivial vector space, containing the zero vector and nothing else.(4 votes)

- Is an n dimensional mean that one or several imaginary numbers are used as a coordinate/coordinates?(2 votes)
- I am not sure if you are familiar with these concepts, but we say that a vector space has dimension n, or is n-dimensional, if it has a basis of n elements.(3 votes)

## Video transcript

When you get into
higher mathematics, you might see a
professor write something like this on a board,
where it's just this R with this extra
backbone right over here. And maybe they write R2. Or if you're looking
at it in a book, it might just be
a bolded capital R with a 2
superscript like this. And if you see this,
they're referring to the two-dimensional
real coordinate space, which sounds very fancy. But one way to
think about it, it's really just the
two-dimensional space that you're used to dealing
with in your coordinate plane. To go a little
bit more abstract, this isn't necessarily
this visual representation. This visual
representation is one way to think about this
real coordinate space. If we were to think about it
a little bit more abstractly, the real R2, the two-dimensional
real coordinate space-- let me write this down-- and the
two-dimensional real coordinate space. And just to break
down the notation, the 2 tells us how many
dimensions we're dealing with, and then the R tells us this
is a real coordinate space. The two-dimensional
real coordinate space is all the possible
real-valued 2-tuples. Let me write that down. This is all possible
real-valued 2-tuples. So what is a 2-tuple? Well, a tuple is an
ordered list of numbers. And since we're talking
about real values, it's going to be ordered
list of real numbers, and a 2-tuple just says it's
an ordered list of 2 numbers. So this is an ordered list
of 2 real-valued numbers. Well, that's exactly
what we did here when we thought about a
two-dimensional vector. This right over
here is a 2-tuple, and this is a
real-valued 2-tuple. Neither of these have
any imaginary parts. So you have a 3 and a 4. Order matters. We view this as a different
2-tuple than, say, 4, 3. And even if we were trying
to represent them in our axes right over here,
this vector, 4, 3, would be 4 along
the horizontal axis, and then 3 along
the vertical axis. And so it would look
something like this. And remember, we don't have
to draw it just over there. We just care about its
magnitude and direction. We could draw it
right over here. This would also be 4, 3,
the column vector, 4 3. So when we're talking
about R2, we're talking about all of the
possible real-valued 2-tuples. So all the possible
vectors that you can have, where each of its components--
and the components are these numbers
right over here-- where each of its components
are a real number. So you might have 3, 4. You could have
negative 3, negative 4. So that would be 1,
2, 3, 1, 2, 3, 4, might look something
like-- actually, I should make the scale
a little bit bigger, so it looks the
same-- 1, 2, 3, 4. So it might look
something like that. So that would be the
vector, negative 3-- let me write a little bit
better than that-- negative 3, and negative 4. So if you were to take all
of the possible 2-tuples, including the vector 0,
0-- so it has no magnitude, and you could debate what its
direction is right over there-- you take all of those
combined, and then you have created your
two-dimensional real coordinate space. And that is referred to as R2. Now, as you can
imagine, the fact that we wrote this
2 here-- we had to specify-- it's like, hey,
well, could I put a 3 there? And I would say, absolutely,
you could put a 3 there. So R3 would be the
three-dimensional real coordinate space. So 3D real coordinate space. And so you would
view this as all the possible
real-valued 3-tuples. So, for example, that
would be a member of R3. And let me actually
label these vectors just so we get in
the habit of it. So let's say we
call this vector x. Let's say we have a vector
b, that looks like this. Negative 1, 5, 3. Both of these would
be members of R3. And if you want to see some
fancy notation, a member of a set-- so this
is a member of R3-- it is a real-valued 3-tuple. Now you say, well, what
would not be a member of R3? Well, this right over
here isn't a 3-tuple. This right over here
is a member of R2. Now, you might be able
to extend it in some way, add a zero or
something, but formally, this is not a 3-tuple. Another thing that would not
be a member of R3-- let's say someone wanted to make
some type of vector that had some imaginary parts in it. So let's say it had i, 0, 1. This is no longer real-valued. We have put an imaginary--
this number up here has an imaginary part. So this is no longer
a real-valued 3-tuple. And what's neat about
linear algebra is, we don't have to stop there. R3 we can visualize, we
can plot these things. In your previous
mathematical career, especially if you have some
type of a hologram or something, it's not hard to visualize
things in three dimensions. But what's neat is that we
can keep extending this. We can go into 4, 5, 6,
7, 20, 100 dimensions. And obviously there it becomes
much harder, if not impossible, to visualize it. But then we can at least
represent it mathematically with an n-tuple of vectors. And so if we were to talk
about a real coordinate space generally, you'll often
see the notation Rn, with n as a superscript. So this right over here is an
n-dimensional real coordinate space.