Hey folks, my name is Nathan Johnston and welcome to lecture 12 for Advanced Linear Algebra. In this class what we're going to do is we're going to look at how to change the basis of a linear transformation once you've represented it as a standard matrix. So representing a linear transformation as a standard matrix, that was a topic that we covered a couple lectures ago, so make sure that you're familiar with that. And now we're going to look at how to change the basis.
So if you create a matrix of a linear transformation using one basis, but then change your mind and decide, oh, I want to use this other basis instead. How do you do that? Okay, and we already talked about how to do this for coordinate vectors of vectors, right?
If you have a vector v and you construct a coordinate vector v with respect to b out of it, well, how do you change that to v with respect to some other basis c? Well, you multiply it by a change of basis matrix. Okay, well, the idea with linear transformations and standard matrices of linear transformations is the exact same, really.
It's just... now you've got an input side and an output side of the matrix or linear transformation. So all that changes is you're going to have to multiply on the right and on the left by change of basis matrices. Okay, so that's going to be our main theorem for today's class. It's going to show us how to change basis for standard matrices of linear transformations, and we're going to see that everything just falls out of work that we did earlier.
We're just stitching together pieces of stuff that we already took care of the nasty details of in previous lectures. Okay, so let's see how this works. Okay, so let's start off with our big beast of a theorem that gets us going for today's lecture. The setup is the same as it has been all of these recent lectures.
Okay, we've got some finite dimensional vector spaces, V and W, finite dimensional, so that we can construct standard matrices in the first place. And then we've got some linear transformation T going between those finite dimensional vector spaces. And then we've got some bases. Okay, and here we're gonna have a whole bunch of bases because we've got sort of two old bases, an old input basis and an old output basis.
And then we've got two new bases, a new input basis and a new output basis. Okay, so you've got two bases B and C of the input space, and you've also got two bases D and E of the output space. Okay, and what this theorem tells us how to do is it tells us how to change. If we already know the standard matrix with respect to the old bases B and D, then you use this formula to construct the standard matrix with respect to the new bases C and E. Okay, of course, you could construct this standard matrix here just via the definition as well, but this formula is kind of nice because it lets you reuse work if you've already constructed this standard matrix ahead of time.
So what is sort of the meat of this theorem? Well it tells you that you take this standard matrix with respect to the old bases and you just sort of pre-and post-multiply by appropriate change of basis matrices to get this new standard matrix. And if you just sort of pay attention to the notation, everything makes sense here, right? What's happening on the right is this matrix changes basis C into B, then this matrix changes B into D while applying the linear transformation T, and then this matrix changes the basis D into E, okay?
So if you sort of just trace all of that through, what you've done is you've transformed basis C into E and applied T. And that's exactly what this standard matrix on the left does. So of course they're the same, okay? But we'll go through a proper proof as well.
and maybe let's just write everything that I just said there in a diagram. So here's how it works. If you start off in some vector space V and you have a linear transformation that maps that vector space into another vector space W, there are all sorts of ways that we can represent these input and output vector spaces via coordinate vectors.
For example, on V we have these two bases B and C, so we could represent it via coordinate vectors with respect to the basis B, or we could represent it via coordinate vectors with respect to the basis C. And if we do that, we're going to get, you know, either way, we're going to get a copy of Rn. We're going to get vectors in n-dimensional space, right, if these bases have n members. But they're going to look kind of different, right?
Like over here, if we represent things in basis B, the coordinate vectors are going to look different than if we used basis C. Okay, and similarly on the output space, we have two bases D and E, both with, say, m members. So we get two different copies of Rm. Okay, but if we use basis D to represent these vectors as coordinate vectors, they're going to look different than if we use basis E.
Okay, so we get sort of two different copies of our m here. Okay, and then after we've represented things as coordinate vectors, we have all sorts of objects for transforming between all the things in this bottom row here. For example, if we want to turn a coordinate vector with respect to basis C into a coordinate vector with respect to basis B, well we multiply by p of, well sort of read this as b from c maybe, okay?
It changes basis c into basis b, right? And similarly if we want to convert coordinate vectors with respect to d into coordinate vectors with respect to e, what do you do? Well you multiply by the change of basis matrix.
This matrix turns coordinate vectors with respect to D into coordinate vectors with respect to E. And we also know how to convert coordinate vectors into linear transformation applied to vector and then coordinate vectors. So we know how to go from V with respect to B into TV with respect to D.
You just multiply by standard matrix, right? That was a theorem from a couple of lectures ago. Okay.
What this theorem that we just looked at says, this theorem right up here, 3.3, it says that, yeah, okay. If you want to go all the way from the bottom right over to the bottom left, there are two things that you could do. If you're converting a coordinate vector with respect to c into a coordinate vector of a linear transformation of a vector with respect to e, well that just sort of almost by definition is just the standard matrix e from c. Okay so that's the matrix they have to multiply by to go from the right over to the left, but you can just trace through the diagram if you multiply by this matrix and then this matrix and then this matrix. you get the exact same thing.
Okay, so this matrix, the standard matrix, must equal the product of those three matrices. Okay, it's just sort of a way of breaking it down into steps. All right, so that's sort of the diagrammatic proof.
Let's go through the algebraic proof now, I guess. And it's just basically the same thing but written out in symbols. Okay, so what we're going to do is we're going to look at this matrix, this product of three matrices, and just multiply it by an arbitrary coordinate vector vc and see what happens.
Okay, so I'm taking that so a triple product there and multiplying by vc and if I do that just focus on one piece at a time so I'm going to focus on this p of b from c times vc here okay and our theorem about change of basis matrices why we cared about these matrices is this is going to return back just v with respect to b right okay Next step, use our theorem about standard matrices. What's this going to do? Well, it's going to apply T to V, and it's going to change the basis to D.
So this is just going to equal T of V with respect to D. Great. And then last piece, now just do the product of these two things, this matrix and this vector. And well, it's just going to change the basis from D to E. So this whole junk here is just T of V with respect to E.
Okay. But... What if, on the other hand, we compute this product? If we compute the standard matrix C into E of T times V with respect to C?
Well then, again, our theorem about standard matrices tells us that this would apply T to V, and also it would change the basis from C into E. Okay, so now we have two different products here. We have this one and this one, and they both give us the same thing. They give us T of V with respect to E. All right.
Well, if this is true for all v and this is true for all v, right, it doesn't matter what vector is over here on the right hand side, we always get the same product, that means this matrix must equal this matrix. We used the same trick last class, right, if you have a matrix times v equals some other matrix times v, no matter what v is, they're the same matrix, all right, so then you just look at what is this matrix, what is this matrix, and they're the same as each other, and that's exactly the theorem. Okay, so That's the theorem and hopefully we've sort of motivated it well enough now. Let's do an example to see how this actually works, to see what sort of stuff we can do with it. Okay, so what we're going to do now is we're going to compute the standard matrix of the transpose map with respect to this sort of weird basis here.
Okay, so first off before we do this, remember a couple lectures ago we computed the standard matrix of the transpose map with respect to the standard basis. Okay, with respect to the basis where every matrix just has a one in a single entry and all the other entries are equal to zero. Alright, so we already have one standard matrix of the transposed map.
I want a different one though, okay, for reasons that we'll get to as we finish up this example. We'll see that it's actually kind of nice to represent the transposed map with respect to this slightly weirder looking basis. This, I mean, as a side note, if you've taken physics courses maybe you've seen these matrices before. They're called the Pauli matrices, okay, and they come up all the time in quantum mechanics, but anyway, for our purposes they're just sort of a mathematically nice basis. Alright, So yeah, what we've already seen is we've seen that the transpose map with respect to the standard basis is just this matrix here, okay?
Almost the identity except things are sort of swapped around in the middle there. Okay, well how do we construct the standard matrix with respect to this weird basis B, this poly basis? Well we could do it by definition, that's fine and dandy, but I'm going to do it a different way.
I'm going to do it using that theorem that we just saw, because we can just change the basis of this matrix. Alright, so that theorem that we just saw, theorem 3.3, says that the standard matrix of T with respect to this new basis, you just have to take the old standard matrix and put appropriate change of basis matrices on the left and right hand side. Okay and you just sort of look at it and make sure that the notation matches up, right? E's, adjacent E's are matching up and you're left with B's on those sides.
Okay so all we have to do is we have to compute these change of basis matrices. Okay so let's start off with p, the change of basis matrix with e from b. Okay, so this one down here.
And remember, whenever you're converting into the standard basis, you can just read off entries and those get your coordinate vectors, which are your columns, okay? So the way that we constructed this matrix here, we're converting into the standard matrix, sorry, into the standard basis, all right? So what you do is you look at these, the members of the old basis b one at a time.
and you just read off the entries 1 0 0 1 1 0 0 1 so that becomes your first column all right and then you look at the next member of b the old basis 0 1 1 0 0 1 1 0 that becomes your second column then you look at the third number of your old basis 0 minus i i 0 0 minus i i 0 that becomes your third column and then the fourth one is 1 0 0 minus 1 1 0 0 minus 1 that becomes your last column So go back to last week's notes if you're having a little trouble doing that computation. We talked about how to compute change of basis matrices then. So we've got one of the two change of basis matrices. And fortunately, the other one, it's just converting the bases the other way around. And we had a theorem that said, hey, if you want to swap the order in which you're changing the bases, you just invert the matrix.
P of B from E, that's just the inverse of P E from B. It's just swapping the bases in the other order. So you just... invert this matrix using the usual Gauss-Jordan elimination method, and you get this matrix down here.
All right, so that gets me the two change of basis matrices that I need. Now, to compute the new standard matrix that I want, t with respect to the weird basis, all you have to do is multiply, right? Change of basis matrix times the old standard matrix times change of basis matrix, and here are the three matrices that we already have.
Just put them down there. do the ugly matrix multiplication. I mean, there's a bit of work to do here, but also there's lots of zeros, so it's not that bad. And at the end of the day, you get this matrix here. So that is the standard matrix of T with respect to that poly basis up above.
Okay, there are a couple notes that I want to make about this final answer here. So first off, if you computed it directly from the definition T with respect to B, you would get the exact same thing. And you can sort of see that here, right? You only got diagonal entries here, everything else is zero. And if you look back at these basis members, what happens when you transpose them?
Well, this guy stays unchanged, this guy stays unchanged, the fourth guy stays unchanged, and this guy, its transpose is exactly equal to its negative. Right, so when you write out coordinate vectors for the transposes of these guys, you're just going to get 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, minus 1, 0, and 0, 0, 0, 1, right? You're just going to, like the linear combinations are all very simple because these guys are all, after you transpose them, just multiples of each other. And those multiples are 1, 1, 1, 1, minus 1, and 1. Okay, so that's sort of where that diagonal shape is coming from. And also, this sort of highlights why it's nice to change basis in the first place sometimes, okay?
Even though it's really tempting to just work with the standard basis whenever you can, sometimes other uglier bases lead to simpler matrix representations, and a diagonal matrix representation is just about the simplest one that you could ask for. So maybe it's nicer to represent the transpose map in this poly basis rather than the standard basis. Okay.
So that'll do it for lecture 12 and for week three. So I will see you in the next video when we start talking about properties of linear transformations, like their range and null space and eigenvalues and all that sort of stuff.