Transformation matrix with respect to a basis

Let's say I've got some linear transformation T that is a mapping from Rn to Rn. So if this is its domain, which is just Rn, then its codomain is also Rn. If you give me some vector in our domain, let's call that vector x, then T will map it to some other member of Rn, which is also the codomain. So it'll map it over here. We could call that the mapping of T, or the mapping of x, or T of x. Since T is a linear transformation, we know that the mapping of x to its codomain is equivalent to x being multiplied by some matrix A. So we know that this thing right here is equal to some matrix A times x. You've seen all of this multiple, multiple times. Just to make sure we understand the wording properly, we said we've used the word that A is the-- we could either call it the matrix for T, or let's say it's the transformation matrix for T. Now, in the last couple of videos, we've learned that the same vector can be represented in different ways. It can be represented in different coordinate systems. When I just write the vector x like that, we just assume that it's being represented in standard coordinates, or it's being represented with respect to the standard basis. So let's be a little bit more particular. This A is the transformation for T only when x is represented in standard coordinates, or only when x is written in coordinates with respect to the standard basis. So let me write a little qualifier here. A is the transformation matrix for T with respect to the standard basis. I never wrote this blue part before. I never even said this blue part before, because the only coordinate system we were dealing with was the standard coordinate system or the coordinates with respect to the standard basis. But now we know that there are multiple coordinate systems. There are multiple ways to represent this vector. There are multiple ways to represent that vector, because Rn has multiple spanning bases. There are multiple bases that can represent Rn, and each of those bases can generate a coordinate system where you can represent any vector in Rn with coordinates with respect to any of those bases. So that last part I said was a bit of a mouthful, so let me make it a little bit more concrete. Let's say that I have some basis B that's made up of n-- it has to be linearly independent. That's the definition of a basis-- of n vectors v1, v2, all the way to vn. Now, these are n linearly independent vectors. Each of these are members of Rn. So B is a basis for Rn, which is just another way of saying that all of these vectors are linearly independent and any vector in Rn can be represented as a linear combination of these guys, which is another way of saying that any vector in Rn can be represented with coordinates with respect to this basis right there. So the same vector x, I'm going to put the same dot here. When we represent it in standard coordinates, it's just going to be that right there, that vector x. But what if we want to represent it in coordinates with respect to this new basis? Well, then that same vector x will look like this. We would denote it by this. The same vector can be represented with respect to this basis. This could be some set of coordinates. This would be some other set of coordinates, but it's still representing the same basis. Likewise, this vector right here, that vector right there, is also in Rn. So it can be represented by some linear combination of these guys, or you can represent it with coordinates with respect to this basis. So that same point right there, I could represent it. So that point is this. But I could represent it with coordinates with respect to my basis just like that. So this is an interesting question. This should maybe bring an interesting question into your brain. If I start off with something that's in standard coordinates, and I apply the transformation T-- that's like applying this matrix A to it or multiplying that thing in standard coordinates times the matrix A-- I then get the mapping of T in standard coordinates. Now, what if I start off with that thing in nonstandard coordinates if I have coordinates with respect to this other basis here? Well, T should still map it to this guy. The transformation, no matter what, should always map from that dot to that dot. It shouldn't care what your coordinates are. So T should still map to that same exact point. T should still be a linear transformation. It can map from x to T of x, but that's the same thing as mapping from this kind of way of labeling x to this way of labeling x. So we could say maybe this guy right here could be some other matrix times this guy over here. So let me write this over here. These are just different coordinate systems. I shouldn't just even say maybe. This guy should be able to be represented. So if I represent the mapping of x in our codomain in coordinates with respect to B-- so that's what that guy is right there-- so if I want to represent that dot with this other coordinate system, coordinates with respect to this basis, it should be equal to the product of some other matrix. Let me call that other matrix D. Some other matrix D times this representation of x times the coordinates of x with respect to my alternate nonstandard coordinate system. I should be able to find some matrix D that does this. Then we would say that D is the transformation matrix for T. A assumes that you have x in terms of standard coordinates. Now D assumes that you have x in coordinates with respect to this basis, so with respect to the basis B. There's no reason why we shouldn't be able to do this. These things are just different ways of representing the exact same vector, the exact same dot in our sets here. So if I represent it one way, the standard way, I multiply by A, and I get Ax. If I represent it in nonstandard coordinates, I should be able to multiply it by some other matrix and get another nonstandard coordinate representation of what it gets mapped to. So let's see if we can find some relation between D and between A. So we learned a couple of videos ago that there's a change of basis matrix that we can generate from this basis. It's pretty easy to generate. The change of basis matrix is just a matrix whose columns are these basis vectors, so v1, v2-- I shouldn't put a comma there. These are just the columns-- v2 all the way to vn. This is an n-by-n matrix. Each of these guys are members of Rn and we have n of them. This is an n-by-n matrix where all of the columns are linearly independent, so we know that C is invertible. These are column vectors right here. So we know that C is invertible. We learned in the last two or three videos that if we have some vector x, and it's being represented by coordinates with respect to our basis B, we can just multiply that by C, and we'll get our vector x. This essentially will tell us the linear combination of these guys that'll get us x. Since C is invertible, we also saw that if we have the standard format for x, or the standard coordinates for x, we can multiply that by C inverse. That will get us the coordinates for x with respect to the basis B. These two things, if you just multiply both sides of this equation-- let me do it in a different color-- if you just multiply both sides of this equation by C inverse on the left-hand side, you're going to get this equation right there. Now given that, let's see if we can find some type of relation. Let's see what D times xB is equal to. So let's say if we take D times xB, so this thing right here should be equal to D times the representation or the coordinates of x with respect to the basis B. That's what we're claiming. We're saying that this guy is equal to D times the representation of x with respect to the coordinates with respect to the basis B. Let me write all of this down. I'll do it right here, because I think it's nice to have this graphic up here. So we can say that D times xB is equal to this thing right here. It's the same thing as the transformation of x represented in coordinates with respect to B, or in these nonstandard coordinates. So it's equal to the transformation of x represented in this coordinate system, represented in coordinates with respect to B. We see that right there. But what is the transformation of x? That's the same thing as A times x. That's kind of the standard transformation if x was represented in standard coordinates. So this is equal to x in standard coordinates times the matrix A. Then that will get us to this dot in standard coordinates, but then we want to convert it to these nonstandard coordinates just like that. Now, if we have this, how can we just figure out what the vector Ax should look like? What this vector should look like? Well, we can look at this equation right here. We have this. This is the same thing as this. Actually. we want to go the other way. We have this. We have that right there. That's this right there. We want to get just this dot represented in regular standard coordinates. So what do we do? We multiply it by C. Let me write it this way. If we multiply both sides of this equation times C, what do we get? We get this right here. Actually, no. I was looking at the right equation the first time. We have this right here, which is the same-- first intuition is always right. We have this, which is the same thing as this right here. So this can be rewritten. This thing can be rewritten as C inverse-- we don't have an x here. We have an Ax here, so C inverse times Ax. The vector Ax represented in these nonstandard coordinates is the same thing as multiplying the inverse of our change of basis matrix times the vector Ax. If I have my vector Ax and I multiply it times the inverse of the change of basis matrix, I will then have a representation of the vector Ax in my nonstandard basis. Now, what is the vector x equal to? Well, the vector x is equal to our change of basis matrix times x represented in these nonstandard coordinates. So this is going to be equal to C inverse A times x. x is just the same thing as C. x is just C times our nonstandard coordinates for x, just like that. Let me summarize it, just because I waffled a little bit on this point right there just because I got a little bit confused. If I start off with the nonstandard representation of x, or x in coordinates with respect to B, I multiply them times D. So if I start with this, I multiply them times D, I get to that point right there. So this right there is the same thing as this point right there. That point right there should be the nonstandard representation of the transformation of x, or the coordinates of the transformation of x with respect to B. Now, the transformation of x, if x is in standard coordinates, is just A times x. So this is just A times x. But I want to represent it in these nonstandard coordinates. Now, A times x in nonstandard coordinates is the same thing as C inverse times A times x, if you think this is the same thing as this. So if you have this and you want to represent it in nonstandard coordinates, you multiply it by C inverse, so then you'll get that representation in nonstandard coordinates. Then finally, we say look, x is the same thing as C times the nonstandard coordinate representation of x. So we can replace x with that right there. So the big takeaway here is that D times the coordinates of x with respect to the basis B is equal to C inverse A times C times the coordinates of x with respect to the basis B. So D must be equal to C inverse AC. So if D is the transformation matrix for T with respect to the basis B-- and let me write here-- and C is the change of basis matrix for B-- let me write that down, might as well because this is our big takeaway-- and A is the transformation-- I'll write it in shorthand-- matrix for T with respect to the standard basis, then we can say-- this is the big takeaway-- that D, our matrix D, is equal to C inverse times A times C. That's our big takeaway from this video, which is really interesting. I don't want you to lose this point. We now understand that A is just for a certain set of coordinates. But there's arbitrary different bases that we can use to represent Rn, so we can have different matrices that represent the linear transformation under different coordinate systems. If we want to figure out those different matrices for different coordinate systems, we can essentially just construct the change of basis matrix for the coordinate system we care about, and then generate our new transformation matrix with respect to the new basis by just applying this result.