We've seen that the concept of the basis is really important in the study of vector spaces. If you have a finite basis, then the dimension of a vector space is the size of the basis. And the really beautiful fact is that all bases must have the same size.
same size in a given vector space. So this is how we define dimension in a meaningful way. It's really awesome. So take for example, I don't know, R3, right?
That's three-dimensional because well just think about E1, E2 and E3, right? The standard basis. That's of size 3, it's linearly independent, spanning, does all the things we need it to do. So it's a really fabulous definition, this concept of dimension, because it makes crisp and precise what before had been kind of vague. The idea of dimension, before studying this, it's always kind of a fuzzy idea in your mind.
But now we have this incredibly crisp definition. What I want to show you today is that a basis allows you to to really translate the abstract into the concrete and really to do this I think we'll start with just an example before we do anything though I want to talk about what a coordinate system is just very vaguely a coordinate system kind of in any situation is some way to describe positions in a space using numbers right Let's space using numbers. The best example I can think of is Cartesian coordinates, as we're going to describe in a second. Cartesian coordinates is where you just have the usual business. You have the xy plane and you describe a position in the xy plane using two numbers.
One to describe the horizontal and one to describe the vertical. So let's take that as an example, shall we, as a starting point. So here's an instructive example.
So let's take my space as R2, right? And let's think about what ordinary Cartesian coordinates look like. So standard Cartesian coordinates.
You'll see why I'm calling them standard in a second. But what are standard Cartesian coordinates? Well, let's take the xy plane and let's take this point here, v, underline, right, that position there.
Now, standard Cartesian coordinates is what? I would describe that using kind of two numbers, right? Whatever this length is and whatever this length is.
So in standard Cartesian coordinates it's 4, 3. Let's think a little bit about how Cartesian... coordinates are constructed though, right? Cartesian coordinates fundamentally are built using two vectors, right? E1 and E2. We are essentially describing the position in the space using those two vectors.
This V here, let's think about what it means in kind of standard coordinates. V is the linear combination of four copies of E1 plus three copies of E2, right? So it's 4 in that direction, and then 3 in that direction to get to V. Okay, so that would mean, I suppose, if we're going to use usual terminology, V is, I would write it as 4 through... in coordinates given by E1 and E2.
It's kind of a funny way to think about it, but it's kind of exactly what coordinates mean when we're doing Cartesian coordinates. Coordinates. Maybe I should say in the coordinate system.
given by this pair of vectors e1 and e2. So really when I say standard Cartesian coordinates, I mean this is a way to describe positions in the plane using the standard basis, right? So this is kind of constructed.
using the standard basis for R2. So when we talk about this position V... as 4,3.
Really what we're saying is it's given by the linear combination 4 copies of E1 plus 3 copies of E2. Okay, and that's kind of what standard Cartesian coordinates are. What would non-standard Cartesian coordinates mean?
Well, I think it should be something where we're still using two numbers to describe a position, but maybe we don't use the standard basis. non-standard okay so let's choose a basis I'm gonna basically come up with a construction like above but without using the standard basis and use a different basis so let's construct this Constructed using... So let's use a different basis.
Let's do 1, 2, 3, 1. It's a linearly independent set of size 2 in R2. It's a basis. This is a non-standard basis.
Non-standard because it's literally not E1, E2. Now, if I gave you these two vectors... I could basically draw a grid in exactly the same manner as this, but it wouldn't be composed of squares like it is here.
It would be composed, well I think, of parallelograms whose edges would be dictated by these two vectors. So in particular, if I draw this, let me show you what happens. I've got exactly the same basic image.
But now, my two basis elements are different, aren't they? This is... I should probably change the order of these if it makes sense.
This is a more sensible... Maybe if, no, it doesn't matter, it's totally fine each way. But I think I'm going to do them like that. The reason I'm doing them like that is because they're in an orientation that you're more used to.
That one there is like the first element, and that one there is like the second one too. Okay, the vector v is in the same position as above. It's exactly the same position, but if I was to try to express this, In this kind of non-standard coordinate system, this kind of warped coordinate system, things will be a bit different, right? So here, I mean, what am I seeing here?
I'm seeing that that is at that kind of end of this parallelogram here. So v is equal to one copy of 3, 1 plus one copy of 1, 2. So I suppose, being a little bit vague here, but it would make sense wouldn't it, is v is, well,, not in the standard coordinates, but in coordinates given by, in coordinate system given by. I'm going to follow the same pattern, thinking about where I sit.
So for example, this location here. What would that mean in this coordinate system? Well, it would be 0 of this plus 2 of this.
It would be 0, 2. This location here. That would be 1 of this minus 1 of this. It would be 1 comma minus 1. So the thing to think about this is the positions in your space are kind of absolute. That V and that V are the same regardless of what coordinate system I choose. But if I'm choosing this as my kind of foundation for the coordinate system versus this, the numbers I associate to this position will be different.
This is kind of a really neat idea, right? You're very used to always thinking in Cartesian coordinates like this, but there will be occasions, as we're going to see, that thinking in slightly non-standard coordinates can really illuminate things greatly. Okay, so let's try to make this general.
Let's think about what's going on here. Basically, to describe a position in the plane in this manner, what I need is a basis. If I have a basis, it's going to give me this grid, basically, and positions will be the same. be dictated by how far along each you've got to go right this is one one because it's one along there and one up there whereas here this is four along here and three up there right so it's just a different way of labeling the positions the same v but it's got different numbers associated to it because it's a different system essentially a different decision about how we describe positions so definition Alright, so what's the essence of this? Well we have a vector space and we need a basis.
So let's take beta is equal to a basis, b1 up to bn, a basis for v, some vector space. And let's choose an element, v and v. Right, what do I know? Well, I'm a basis. That means I span v, which means that v is a linear combination of these.
That's the first thing. So this implies that v is equal to lambda1 b1 plus dot dot dot lambda n bn. So that's the spanning.
It turns out the following is true. The linear independence... says that this is actually uniquely determined by v. So the coefficients, or the weights, they're unique.
You can't express v in two different ways. This implies the lambda i are unique. You need to be. There's not two different ways of doing it, because if you had two different ways of doing it, you could take the difference, and that would give you a linear dependence.
Okay, so here's the definition. The definition is, I'm going to write v sub b to be equal to the column vector of real numbers given by these weights. So this is me expressing v in this coordinate system.
So this is v in b coordinates, or beta coordinates. And this, of course, is just a vector in Rn. This is a very important definition. And in many ways we're going to see it's going to be what saves us from this incredible level of abstraction we've been working in. Because it's going to allow us to take abstract things, whatever they might be, and convert them into very concrete things.
So let's do an example just to make sure we understand this. Some examples. Okay, first one is we've just done this exact procedure. If I have v is equal to r2, beta is equal to, the vector is equal to 3, 1, 1, 2. that basis.
This implied, well, above our v was in standard coordinates, that's how we'd always describe it, but if I want to express this in beta coordinates, this is rather confusing, it's. Okay. So this is what we did before and that's because of the fact 4, 3 is the linear combination of these two or add them together right 1 times that plus 1 times that is 4, 3 so that's always a bit confusing it's like all the different it's the same point right it's just you're expressing a different coordinate system real important example of course is the one coming from the standard basis the first one we looked at so I'm gonna always write the standard basis from now on with this kind of squiggly e thing this is going to be equal to e1 dot dot dot up to en standard basis So this is our kind of absolute fundamental one.
If I take v equal to just lambda 1 dot dot dot lambda n, let's do a little sanity check. What would happen if I try to write this in standard coordinates? Well by definition on the standard basis v is equal to lambda 1 e1 plus dot dot dot lambda and en it's the really nice thing about the standard basis you can immediately write down anything in here as a linear combination just immediately you're being told the weight so this tells you lambda one dot dot dot to lambda n in standard coordinates it's just it again right thank goodness we'd have come up with a dreadful definition if that wasn't the case so really the interest for us at least in rn is choosing non-standard coordinate systems because they may make vectors look quite different, okay, they may make things much more transparent. That's going to hint to the future, we're going to see that making very clever choices of coordinate systems or bases will make linear transformations much more, much more kind of transparent in their behavior. Okay, so in order to express something in coordinates, the aim of the game is to write it as a linear combination, right?
That's what this is. Where basically this information is identical to this information. If I know what your basis is and what these numbers are, boom, I've got v because I just write this down.
If v is equal to Rn, this is a very standard computation. Computation when v is equal to Rn. So again, it seems a bit weird. Why would you bother doing this with a non-standard basis?
As I said, we're going to see. But if beta is equal to b1 dot dot dot, that's just a collection of column vectors, remember, in Rn, a basis for Rn, then the following is true. How do I express some v as a linear combination? You solve a linear system.
So if you're looking to write v, if you're looking to write v as a... Or in coordinates given by b, what do you do? Well, you just do row reduction. Turns out that gives you the exact thing.
It's really nice. So if you take the time to do this, I should say beta is a basis. So when I write the columns down, it's got to be an invertible matrix. So you can always... Do row reduction to put this in the form, the identity, n by n matrix.
And what are you left with? Well, you are left with v and b coordinates. It's a really nice fact. This is row reduction. This process only really makes sense if you're dealing with v is equal to rn, right?
Because for a general vector space, I mean... I mean, v is just v. The b's, the betas, are not going to be column vectors, so you can't do this. But if you want a slick way to write down a vector in non-standard coordinates, this is what you do. It's really clean.
We'll see an example in a little bit. I do want to show an example where we're not dealing with Euclidean space. So here's a non...
Non-Euclidean example. When I say non-Euclidean, I mean we're not in our end. The simplest example I can think of is polynomials with some fixed upper degree. So, we saw this last time. That collection is really the standard basis.
in polynomials of degree at most n with real coefficients. And if I take v, remember v is an element of this vector space, it's just a polynomial, it's going to be given by this, where these are just the coefficients, I mean this is the meaning of a polynomial basically. Well then it's pretty obvious basically this is me writing v as a linear combination of these.
These are the weights. So if I was to use this basis to represent this polynomial What is it? It's just the column vector lambda 0 lambda 1 dot dot dot down to lambda m. It starts with lambda 0, right?
I've deliberately done 0 there instead of lambda 1 just because it would be confusing to be out of sync later on. Okay, now... This basically means it doesn't matter how abstract your vector space is, the moment I'm expressing it in coordinates, you're getting something in Euclidean space, right, just columns of real numbers. So this is in R n plus 1. Alright, so here is an observation.
The reason this is so powerful... It's quite abstract, but really crucial. So what are we doing? We're taking a V, vector space V, we're choosing a basis, all very abstract, and I am, in a sense, identifying the vector V with this column of numbers. So this process identifies Your abstract vector space V with Rn, assuming that the basis of size n.
What do I mean by that? Well, this process, I'll write it like this, is actually transformation. It's a recipe, it's a function, which starts in Rn and ends up at Rn.
What does it do? It takes your V, your abstract guy, whatever it might be, and it spits out... V in coordinates, right, which is just some concrete information.
And the beauty of this is, I mean, V and Rn are both linear, linear, or sorry, both vector spaces. This identification is actually linear. It's a linear transformation. It's really nice. It's also one-to-one, and it's onto.
We're really kind of identifying both sides, right? They kind of match up really perfectly, and that's because beta is a basis. So this means if I fix a basis, I can translate things over to Rn.
be unharmed. Any question I might have about a bunch of vectors in V, if I translate them over to RN using a basis, so a specific coordinate system, I can answer the questions over in RN and then I can translate back. So this is a really powerful tool.
It allows us to take the abstract and convert it into the concrete. I will also say the basis here. This basis, it's great, it matches up with the standard basis. So in this identification, E1 corresponds to B1, E2 corresponds to B2. It's really nice.
That's just a really nice thing to know about this identification. It kind of makes sense, right? It matches the standard basis. So as I said, the power of this is it's going to allow us to translate information in one vector space into Euclidean space. And I can do all sorts of stuff in Euclidean space.
I'm used to row reduction, all that kind of business. So it means all of this stuff we've been doing in RN, RM, all this business is actually very, very powerful. It will be able to extend to the more abstract setting if I choose a basis and choose a coordinate system.
So let me make that a little bit more precise facts. V1, let's say to VP, linearly independent in V. If and only if, when I translate to coordinates, linearly independent. So again, if I was confronted with an abstract question about are these vectors linearly independent, if I choose a coordinate system, boom, I have a collection of column vectors in Rn, I can check linear independence by doing row reduction.
It's fabulous. Similarly, V1 to VP, spanning. That means the span linear combinations is all of V. You guessed it, this is...
Maybe I should write this to be a bit cleaner. So the span of these guys is equal to v, if and only if the span of their corresponding coordinates is all of Rn. It's such a great result, right? Something I think is really important to understand is the power of this technique, or these constructions, it translates from the abstract to the concrete. And nice as the abstract is, it is always important we can actually compute things, right?
And we're masters of computing things in Rn, right? So if we can use that, we're in good shape. So here's an exercise, an example. So this is going to look pretty difficult, but let's translate it, right? So does 1 plus x, x minus x squared, 1 plus x squared, span?
P2R. Abstract question. Can I write all vectors in there, all polynomials of a degree less than or equal to 2, as linear combinations of these?
So that's pretty hard, right? What am I going to do? How am I going to approach this?
Well, let's just choose the standard coordinates, 1, x, and x squared. What does that mean? Well, if I'm going to translate this across...
What are these coordinates going to be? It's going to be very straightforward. It's easy to read off.
This is the beauty of this choice of bases. It's what? It's going to be the vectors,, and.
Is that right? Made a mistake? Second one? This guy.
It should be. So... These span, if and only if these span.
So let's check it. How do I check that? Well, I just do standard business.
I write down the associated matrix. And I do row reduction on it. If I do row reduction on it, what's going to happen?
1, 0, 1. Take away the first and the second. 0, 1, 0. Is that right? No, sorry, it should be a minus one.
And then I've got a zero minus one one. Then eliminate that minus one underneath the central one. One zero one. Zero one minus one.
If I do this I get zero zero zero. Boom, I'm not spanning. I've got two pivots, right?
And they are not in every row because we need three of them. So no, not spanning. Great result, right? We've taken this abstract question about polynomials by fixing a basis. I've translated it into the world of Euclidean space, into R3 here, and then it's just a question of doing something we're very familiar with.
wonderful idea, right? It's an incredibly powerful idea. But one must understand its shortcomings. And the major shortcoming of this is potentially it's very arbitrary.
Remember, in a general vector space, there's no kind of one basis. In some vector spaces, there's a basis that kind of jumps out at you, like the standard basis in RN. But... We know that we could do this coordinate system for different bases, and I think it's incredibly important to understand how they're related to each other. It's tempting to say, well, let's just work in standard coordinates and forget the rest, but that won't be a good plan.
Something will be really illuminated, potentially, by working in non-standard coordinates. So the question is, what happens when we change bases? Alright, what do I mean by that? Well, changing bases will change the coordinate system.
You'll get different numbers. Let's just go to our original example, right? This guy. Standard coordinates, that is 4, 3. And non-standard coordinates given by this, it's 1, 1. Those are different, different elements of R2. They're representing the same point, but from different perspectives.
So I think it's going to be really important to understand how we can at least translate between different coordinate systems. And this is a delicate thing. So let's choose two bases.
B1 to Bn and another one let's do capital C. C1 to Cn. Bases. in V. Now two different bases, you get two different coordinate systems. So let's try to visualize this carefully.
We've got V. If I translate into one coordinate system, I'm gonna get Rn of course. Okay that would send my vector V to V written in B coordinates, but at the same time If I translate into... How neatly can I do this?
If I translate into coordinates given by C. It's gonna be different. It's gonna be different. Now, remember, so basically we're trying to work out what's the relationship between these, right?
If I'm in beta coordinates, I'm something. If I'm in C coordinates, I'm something. What's the relationship? Well, first off, this is a 1 to 1 onto map.
That means it's got an inverse. So this can be reversed. If it can be reversed, I could start here, go back to here, and then go into C coordinates.
That would be a composition. And it would be a composition which gave me a map from Rn to Rn. And what would this map do?
Well, it would take something written in beta coordinates and it would send it across to something written in C coordinates. Wouldn't it, right? Because going backwards you'd just get V and then you'd get C. Alright, now this map, let's call it T for a second, that map is a linear transformation.
It's not completely obvious, but it's true. It makes sense because these are both linear. Okay, if that's a linear transformation, that means it corresponds to a matrix. It's from Rn to Rn.
So that means I'm going to get an n by n matrix, which gives me the translation from beta coordinates to C coordinates. Now I've got to work out what it is. What is the standard of T, well, it's equal to the matrix given by, well, what happens to the standard basis. That's why the reason we call it the standard matrix. So we need to work out what these are.
So we need to start down here. I'll write it over here, an EI, and I've got to work out what happens. Well, first of all, we know that EI, the standard basis, matches...
How am I going to do this? That looks nice. It matches with what? With the beta basis.
So this would go up to beta I there. All right? But now if I go back down to C coordinates, what would happen? Going back down into C coordinates would give me, so this is a very tricky piece of logic, bi written C coordinates. So what does that mean?
That means the standard matrix of T, the thing which will translate from beta coordinates to C coordinates, is what it's going to be. The first basis written in C coordinates, well, B1 is the beta written in C coordinates as we go along. Now this matrix, we're going to give it some notation.
Let's write this as, okay, the job of this matrix will be to translate from beta coordinates to C coordinates. So I'm going to write it in a slightly strange way. I'm going to write P. And I'm going to write beta here, and then an arrow, and then c.
So this is notation to say it's going to switch from beta to c coordinates. Notation. And it's called a change of basis matrix.
Now, why is it called that? Let's look at the key property. The key property is the following.
Its job is to translate from beta coordinates to c-coordinates. So if I take v written in beta coordinates, and I multiply it on this side by beta, C like this. You can see why I've drawn it like this now. If I write that, this is V written in C coordinates.
Of course it is, right? It's doing the job of this. So it's going from Rn to Rn, multiplication by the standard matrix.
It takes that to that. This is the key property, and this is for all V. in your vector space V. This matrix translates between coordinate systems. And it's quite a concrete thing, because what do we do?
Well, we just take the beta, and we express each of them in C coordinates. That's just a bunch of vectors in Rn, and write it as a matrix. Now, how do you compute this thing? Well, in general, it's going to be a pretty serious thing, but what about if V was equal to Rn? We saw before that it was basically a row reduction business going on.
In fact, let's go back a second. Computing things in coordinates involved doing this, right? If I wanted to write something in beta coordinates, I wrote it there, and I did the row reduction, and I ended up getting it written in beta coordinates. So what I want to do here, I don't want to write things in beta coordinates. I want to write them in C coordinates.
So I should basically do exactly what we've got above, but put C in place of beta. So if I have exactly the same notation... And don't worry, we will do a specific example in a second.
These are bases in Rn. What would the change of basis matrix be? Well, if I was to write C1 to Cn, then make kind of a super version of the augmented matrix, B1.
to bn. If I do row reduction on this, getting the identity here, what would I be left with? On the other side, I would end up with beta 1 written in c coordinates, dot dot dot, all the way to beta n written in c coordinates.
But what's that equal to? So maybe I should say the following. It's not implies, it's row reduction. So what's this equal to? Ultimately this is equal to the identity and we've got my change of basis matrix.
Right, it's that. It's a really slick way of doing it. So once again, row reduction gives us some really interesting things that we didn't, we had never really considered this before.
But the process of doing row reduction gives me this really nice, really nice thing. I should say, before I do a concrete example, there's a consequence of this which is really nice. I mean, this reminds me very much of inverting a matrix, right? If we think about it, the following is true.
So if I started off with beta1, dot dot dot, to beta n, one standard basis on this side, and I do the standard basis here. Well, that's just the identity matrix. So if I do row reduction, what do you end up getting?
In fact, I should use the same notation. I should use beta, sorry. I should use a C, C1, Cn, sorry.
So if you do row reduction on this, if you do row reduction on this, What do you get? Well, we get the identity matrix, and then this is going to be the inverse, I'll write it like this, inverse of the matrix c1 dot dot dot to cn. Similarly, from the other side, I had the identity, or the standard matrix, or standard basis, sorry.
I was doing this. Well, row reduction, you don't do anything, right? You're already there.
I've got i, n, and it's just going to be beta 1. It's almost too ridiculous to write down, but I'll do it. It's that matrix, right? So both of these are examples of this. It's just I'm choosing the standard basis as one of my two guys. In particular, this now is what?
It's going to be the change of basis matrix from the standard basis to C. So to find that, we just write down C as a matrix and find its inverse, right? This guy, though, is what?
Well, same logic, but kind of reversed. This guy is the change of basis matrix from this basis to the standard basis. So this is really nice things to know in general.
So there's a really simple special case. If you're changing from beta to the standard matrix, the standard basis, the change of basis matrix is just your betas written. So you don't really need to do much, but that's a very special case. Okay, that is super confusing when you first see it. Let's do an example.
Let's put all of this together. Okay, so let's do beta. So first of all, let's do v is equal to r2.
In fact, let's do the example from the beginning. So let's have the standard matrix, e1, e2. And let's have c as this non-standard basis.
3, 1, 2, 1. Sorry, 1, 2. We'll take this, right? And this is... bases in R2.
Okay, and let's take V, the point in standard coordinates, for 3. Okay, so let's go through all of this. First of all, I want to work out the matrix which switches from beta coordinates to C coordinates. Let's just write this down carefully. So first of all, we want to know what is the thing which switches from beta coordinates. to C coordinates.
Well, how do we do this? We write down 3, 1, 1, 2, 1, 0, 0, 1 and I do row reduction. Now doing row reduction takes a bit of time, but you end up getting the following. You end up getting... In fact, let me do this the other way around.
It doesn't really make any difference. In fact, no, I'm going to do this fully. Apologies. So I'll do this properly for us.
I won't skip any steps. Okay, so the first thing I'm going to do is I'm going to flip the two rows. It's going to be 1, 1, 2, 0, 1. So I'm doing this just to make the...
Position there 1, which is simple, rather than dividing through by 3. Then I've got a 3 and a 1, and a 1 and a 0. Okay, so let's do the next one. Let's eliminate the 3. So I'll take 3 of the first row from the second. 1, 2. 0, 1, 0, minus 5, is that correct? Yes.
1 and minus 3 here. So that's me taking 3 away. Is that correct?
I think so. So I'm part of the way there. Now the next, it's a bit ugly. I want to get a 1 there.
I'm going to have to divide through by minus 5. 1, 0, 2. 0, 1. Okay, to get a 1 there, divide through by minus 5, minus a fifth, three-fifths. Next, so what do we have to do next? Next we have to get rid of that 2. This is the last step, I suppose.
So to do that, I'm going to have to take away 2 of the second row from the first, and then I'll have the appropriate format. Okay, oh my goodness. So minus 1 over 5, 3 fifths. Okay, so taking away 2, 2 fifths there. Taking away 2, 1 minus 1 over 5, 3 fifths.
Minus 6 over 5 is minus a fifth. Fun, fun, fun. Okay, so let's get all this correct.
I've done what I need to do. I've computed this, right? That gives me the change of basis matrix.
So this will imply... v beta to c is equal to this matrix, 2 fifths minus 1 fifth, minus 1 fifth, 3 fifths. Okay, so let's do everything now. Remember, v is equal to 4, 3. In standard coordinates, it's just equal to 4, 3 again, of course. In c coordinates, as we saw...
It's that plus that, so it's 1, 1. I'm going to show to you that this actually does translate from the beta coordinates to the C coordinates. So PC, B, V and B coordinates. So this should give me the C coordinates. Let's write this down. So this is 2 fifths minus 1 fifth minus 1 fifth, 3 fifths times 4, 3. So what's this equal to?
This now is equal to 2 fifths times 4, that's 8 fifths, minus 3 fifths, 5 fifths, 1. Minus 4 fifths plus 9 fifths, 5 fifths, 1. Which is v written in c coordinates. Fabulous. It does what we expect it to do. This change of basis business is fiddly. It takes time to absorb and it's really important you don't just blindly memorize this kind of change of basis procedure because you will be given scenarios where you are not necessarily working in Rn, right?
In which case, you really do need to think carefully about the definition. Okay, so this is a tricky thing. It's going to turn up when we start to think about how to associate matrices.
to abstract linear transformations. But yeah, it's a really important idea because fundamentally in a vector space, there's no one basis. So it's kind of an arbitrary choice in many ways what basis you choose to give you a coordinate system.
And you must be able to translate freely between them, especially as we're going to see certain choices of coordinate system will be very good for certain circumstances, whereas perhaps the standard one won't be.