Real coordinate spaces

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.