Change of basis
Coordinates with respect to a basis
Let's say I've got some subspace of Rn. Let's say V is a subspace of Rn. And let's say the set B-- I'll do it in blue-- let's say the set B is a basis for V, so it's got a bunch of vectors in it. And let's say it's got v1, v2, all the way to vk. So you can see we have k vectors, so v is a k-dimensional subspace. So that means that if I have some vector a-- let's say I have some vector a that is a member of my subspace-- that means that I can represent a as a linear combination of these characters right there. So I could write a as being equal to some constant times my first basis vector, plus some other constant, times my second basis vector. And then I can keep going all the way to a kth constant times my k basis vector. Now, I've used the term coordinates fairly loosely in the past. And now we're going to have a more precise definition. I am going to call these constants here c1, c2, all the way through ck. I'm going to call them the-- I'll do it in a new color-- the coordinates of a, with respect to our basis B, with respect to B. And we could also write it like this. We could also write I have my vector a, so I have my vector a. But if I wanted to write my vector a in coordinates with respect to this basis set B, I would write it like this. Put brackets around it, and put the basis set right there. And this says I'm going to write this in coordinates with respect to this basis set. So, I will then write it like this. I'm just going to put these weights there, these constant terms, on the on the linear combination that I have to get of my basis vectors to get a. So there's c1, c2, all the way to ck. Now there's one slightly interesting thing, or maybe very interesting thing, to point out here. V is as basis of Rn, so anything in V is also going to be in Rn. But V has k vectors. It has dimension k. And that k could be as high as n, but it might be something smaller. Maybe we have two vectors in R3, in which case v would be a plane in R3, but we can abstract that to further dimensions. But when you specify something that is in our subspace, with respect to its basis, notice you only have to have that many-- the dimension of your subspaces-- you only have to have that many coordinates. So even though a is a member of Rn, I only have to give it k coordinates. Because, essentially, you're giving it positions within, let's say, if this was a plane, within the plane that is your subspace. Let me make this a little bit more concrete. Let me do some examples. So let's say we have some subspace. Let me clear this out. Let's say I have a couple of vectors. Let's say v1 is the vector 2, 1. And let's say v2 is the vector 1, 2. Now you might immediately see that the basis, or the set, of v1 and v2, this is a basis for R2, which means that any vector in R2 can be represented as a linear combination of these guys. I could do a visual argument. Or we also know that, look, R2 is two-dimensional, and we have two basic vectors right here, and they are linearly independent. You can verify that. In fact, the easiest way to verify that, is if you just take 2, 1 and 1, 2, and you put it in reduced row echelon form, you're going to get the 2 by 2 identity matrix. You're going to get 1, 0, 0, 1, and that lets you know that this guy and this guy are both basis vectors. So, that's all review. We've seen that before. But let's visualize these. Let's visualize these guys. So if I were to just graph it in the way we normally graph these vectors, what does 2, 1 look like? let me draw some axes here. Let me draw it. We'll do it in a different color. Let's say that is my vertical axis, and this is my horizontal axis. And 2, 1 might look like this. We're going to go out 1, 2, and then we're going to go up 1. So that is our vector 1, right there. That is 2, 1, that's our vector 1. And then 1, 2 might look like, or it does look like this, if I draw it in a standard position. 1, then you go up 2. 1, 2 looks like this. So when we talk about coordinates with respect to this basis, let me pick some member of R2. I'll engineer it so that I can easily find a linear combination. Let me take 3 times v1, plus 2 times v2. What is that going to be equal to? That's going to be equal to the vector. So 3 times 2, which is 6, plus 2 times 1. So it's the vector 8, and then I have the vector 3 times v1, plus 2 times that. So 8, 7, right? 3 plus 2 times 2 is for 8, 7. So if we were to just graph 8, 7 in the traditional way, we would go 1, 2, 3, 4, 5, 6, 7, 8, and then we would go up 1, 2, 3, 4, 5, 6, 7. And we would have a vector, I'm not going to draw it out here, but it would specify that point right there. That would be this point. If you view these as coordinates, we would view that as point 8, 7 right there. Maybe I'll write it like that. That's the point 8, 7. If I wanted to draw the spectrum in standard position, I would draw a vector that ends right there. Now, we have this is basis here, this basis B, represented by these two vectors. This is B1 and B2. And what we want to do is represent this guy. Let's say that this is vector-- let me call that vector a-- so vector a is equal to 8, 7. Now, we know that if we wanted to represent vector a as a linear combination of my basis vectors, it's going to be 3 times you v1, plus 2 times v2. So just given what we just saw in the earlier part of this video, we can write that the vector a, with respect to the basis B-- maybe I'll do it in the same color as the basis-- with respect to the basis B, is equal to these weights on the basis vectors, is equal to 3 and 2. And let's see if we can visually understand why this makes sense. We're saying that in some new coordinate system, where this vector can be represented as 3, 2-- and the way you think about the new coordinate system is, in this old coordinate system we hashed out 1's in the horizontal axis, and that was our first coordinate. And we hashed out 1's in the vertical direction. That was our second coordinate. Now in our new system, what's our first coordinate? Our first coordinate is going to be multiples of v1. This is v1 or this is v1, so it's multiples of v1. So that's 1 times v1. Then if we do 2 times v1, we're going to get, over here, 2 times v1 would get us to, what? 2 would get us to 4, 2. 3 times v1 would get us to 6, 3. So let's see, 1, 2, 3, 4, 5, 6, 7, 8. So it's 6 and then 3, just like that. And then 4 times v1 would get us to 8 and 4. So you can imagine that what I'm drawing here, this is kind of the axes, the first-term axis generated by v1. So I could draw it-- let me do it in this blue color-- so you can imagine it like this. This would be a straight line, just like that. And then the coordinate tells me how many v1's do I have. So I would hash off the coordinate system like this. Instead of doing increments of 1, I'm going to do increments of v1. I'll write it just like that. As you go 9, 10, we're going to go up one more to 5, something like that. Now, the second coordinate tells you increments of v2. So this is our first increment of v2, then our second increment if we go to 4, is going to be 4, 2. Just like that. That's going to be 6 and 3. It's going to be just like that. It's going to be 6 and 3. So it's going to look something like this. So if you want to think of it is a bit of a, well, you should be thinking of it as a coordinate system. You can have this new skewed graph paper, where any point you can now specify it as going in the v1 direction by some amount, and then by going in the v2 direction by some amount. Let me draw that as this graph paper. So I could draw another version of v2, just like that, just all the multiples of v2. I could shift them like that. I could do another one like this. I could do another. That one is a little bit not neat enough. I could do it like that. I could do-- I think you're getting the idea. Let me make that a little bit neater. This might have been useful to do with another tool. And then I could do all the multiples of v1 like this. I'm doing a graph paper right here. So it looks something like this. It would look something like this. It would look something like this. And so you can imagine the skewed graph paper, if I did it all over the place with this kind of green and this blue. So in our new coordinate system, we're seeing 3, 2. So that means 3 times our first direction, which happens to be the v1 direction. It's no longer the horizontal direction. It's the v1 direction. We're going 1, 2, and then we go 3, like that. And then we're going to go 2 in the v2 direction. So we're going to go 1, 2 in the v2 direction. And so our point is going to be right there. You could imagine going like this. You go 3 in the v1 direction, and then you go 1, 2 in the v2 direction and you get to our point. Or you could go, kind of in your v2 direction and then your v1 direction, but either way you're going to get to your original point. So that vector, or the position specified by the vector, 8, 7, could just as easily be specified in our new coordinate system by the coordinates 3, 2. Because we're saying 3 times v1, and then, plus 3 times v1. It takes us in this direction. We're going 3 notches in the v1 direction, and then we're going 2 notches in the v2 direction. And so that's why these are called coordinates. You're literally saying how many spaces in the v1 direction to go, and then how many spaces in the v2 direction to go. But this might, I guess, lead you to the obvious question. Why haven't we been using the coordinates before? Like I might have been saying all along. I've been saying all along. Let's say I have some vector lowercase b, that is equal to, I don't know, let's say it's equal to-- I'll do it in an R2, just because it's easy to visualize-- let's say it's equal to 3 minus 1. If we were to graph it it would look something like this. We would go, 1, 2, 3, and then we would go down 1. So it would look something like this. It would specify this point. But why have we been calling 3 and negative 1 coordinates? Why have we been calling 3 and minus 1 coordinates? We've been doing it well before we learned linear algebra. We called these coordinates all the way from when we first learned how to graph. Why are we calling those coordinates? Or how does this meeting of coordinates relate to these coordinates with respect to a basis? Well, these are coordinates with respect to a basis. These are actually coordinates with respect to the standard basis. If you imagine, let's see, the standard basis in R2 looks like this. We could have e1, which is 1, 0, and we have e2, which is 0, 1. This is just the convention for the standard basis in R2. And so we could say s is equal to the set of e1 and e2. Then we say that s is the standard basis for R2. And it's a standard basis because these two guys are orthogonal. This is 1 in the horizontal direction. This is 1 in the vertical direction. And any vector in R2-- let's say I have some vector x, y in R2-- it's going to be equal to x times e1, plus y times e2. So we can say that if you want to write some vector x, y, if you wanted to write with respect to this standard basis right here, it's going to be equal to the coordinates by the definition that we did earlier in this video of the basis vectors right there. Or these weights on our e1's and e2's, so it's going to be equal to, well the weight there is x and the weight here is y. So these coordinates that we've been talking about from the get-go, these are definitely coordinates. They're consistent with our definition of coordinates in this video. But we can maybe be a little bit more precise. We can now call them the coordinates with respect to the standard basis. Or we could call them, we could call these right here the standard coordinates. I just wanted to point this out. This might be almost trivially simple, or a bit obvious. But I just wanted to show that our old usage of the word coordinates was not inconsistent with this new definition of coordinates as being the weights on some basis vectors. Because even in our old coordinates, or the old way we used them, these really were weights on our standard basis vectors.