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Real coordinate spaces

Video transcript

when you get into higher mathematics you might see a professor write something like this on a board whereas this is our with this extra backbone right over here and maybe they write our two or if you're looking at it in a book it might just be a bolded capital R with a two superscript like this and if you see this they are referring to the two dimensional real coordinate space real coordinate space which sounds very fancy real coordinate space 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 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 R to the real the two-dimensional real coordinate space let me write this down so and the two the two dimensional dimensional real coordinate space and just to break down the notation the two tells us how many dimensions we're dealing with and the R tells us this is a real coordinate space the two-dimensional real coordinate space is all the possible real valued two tuples let me write that down this is all possible all possible real valued real valued two tuples so what is a 2-tuple well a tuple a tuple is an ordered list of numbers and since we're talking about real value it's going to be an ordered list of real numbers and a 2-tuple just says it's an ordered list of two numbers so this is an ordered list of two real valued numbers well that's what exactly what we did here when we thought about it two dimensional vector this right over here is a 2-tuple and this is a real value to tuple neither of these have any imaginary parts so you have a three and a four order matters we view this as a different two tuple then say then say four three four three and even if we were try to represent them in in our in our axis right over here this vector for three would be four along the horizontal axis and then three and then three 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 we could draw it right over here that this would also be this would also be four three the column vector four three so when we're talking about our two we're talking about all of the possible real value to tuples so all of the possible 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 three four you could have negative three negative four so that would be one two three one two three four might look something like I haven't actually I should make the scale a little bit bigger that so it looks the same one two three four so it might look something something like that so that would be the vector negative three right a little bit better than that negative three and negative four so if you were to take all of the possible two 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 to here we had to specify its ok well could I put a three 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 coordinate space and so you would view this as all the possible real valued three tuples so real valued three tuples so for example for example that would be a member of r3 and let me let me actually label these vectors just so we get in the habit of it so let's say this vector we call this vector let's say we have a vector B that looks like this negative 1 5 3 both of these would be members of our three and if you want to see some fancy notation a member of a set so this is a member this is a member of our three it is a real-valued three tuple now you say well what would not be a member of r3 well this right over here is in two three tuple this right over here is a member of r2 now you might be able to extend it in some way at a zero or something but formally this is not a three tuple another thing that would not be a member of r3 let's say someone wanted to make some type of some type of vector that had some imaginary parts in it so let's say it had AI 0 1 this is no longer real valued we have put we have put an imaginary this this number up here has a imaginary part so this is no longer a real valued three tuple and what's neat about linear algebra is we don't have to stop there are three we can visualize we can plot these things we've already probably in your 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 four or five six seven twenty a hundred dimensions and obviously there it becomes much harder if not impossible to visualize it but then we can at least represent it mathematically with a sit with a n tuple of vectors and so if we were to talk about a real coordinate space generally you'll often see the notation are n with n is a superscript so this right over here is an n-dimensional n-dimensional real coordinate space real coordinate real coordinate space