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 that is RN and then its codomain is also RN is also R n if you give me some vector in our domain let's call that vector X then T will map it T will map it to some other member of our of RN which is also the co domain so it'll map it over here and we could call that the mapping of T or the mapping of X or T of X and 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 and just to make sure we understand the wording properly we said we've used the word that a is the we could even call it the matrix for T or let's say it's the transformation matrix the trance formation Meishan matrix 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 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 a this a is the transformation for T only when X 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 for T a is the transformation matrix for T with with respect with respect to the standard basis with respect to the standard basis I never wrote this blue book part before I never even said this blue part before because the only 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 basis or there's multiple bases that can represent RN and each of those bases are essentially can generate a coordinate system where you can represent any vector in RN by sum with with with a coordinate with coordinates with respect to any of those bases so that 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 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 as a basis 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 the same dot here when we represent it in standard coordinates we write 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 we could call that same that same vector X will look like this we would give we would we would denote it by this the same vector can be represented with respect to this basis so with you know this be some set of coordinates this would be some other set of coordinates but is 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 if or 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 non-standard coordinates if I have a coordinates with respect to this other basis here well T should still map it to this guy right T should be T should be it should T 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 so let me draw another T should still map to that same exact point T should still be a linear transformation and it could 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 maybe T could be this transfer maybe this guy right here could be some other matrix times this guy over here so let me write this over here so maybe T I mean these are just different coordinate systems so maybe I should I shouldn't just even say maybe this guy should be able to be represented so if I represent the the the mapping of X in our codomain in coordinates with respect to B so that's what that guy is right there's if I want to map represent that dot with this other coordinate system with coordinates with respect to this basis I want to represent it it should be 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 referenced and 'red coordinate system I should be able to find some matrix D that does this and then we would call D we would say that D is the transformation matrix matrix for T and a was with respect to a assumes that you have X in terms in standard coordinates but now D assumes that you have X in this other in in coordinates with respect to this basis so with respect with respect to the basis the basis B 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 it by a and I get ax if I represent it in non-standard coordinates I should be able to multiply it by some other matrix and get another non-standard coordinate representation of what it gets mapped to so let's see if we can find some relation between D between D and between a so we learned a couple of videos ago that there's we can with there's a there's a change of basis matrix that we can generate from this basis and it's pretty easy to generate the change of basis matrix is just a matrix whose columns are these basis vectors so v1 v2 and 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 and where all of the columns are linearly independent so we know that C is in aníbal these are all column vectors right here so we know that c is invertible and we learned in the last two or three videos that that if we have some vector X if we have our vector X and it's being represented by coordinates with respect to our basis B we can just multiply that by C we can multiply that by C and we'll get our vector X this is essentially it'll tell us the linear combination of these guys that'll get us X and since C is invertible we also saw we also saw that if we have if we have just the standard format for X or the standard coordinates for X we can multiply that by C inverse and then that will get us that'll get us the coordinates for X the coordinates for X with respect to B with respect to the basis B and these two things you know if you just multiply both sides of this equation both sides I'm going to 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 now we know or we're saying let's see what D times xB is equal to so let's say we if we take D times xB so this thing right here should be equal to D times D times 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 right so let me write all of this down let's scroll down I'll do it right here because I think it's nice to have this graphic up here so we can say we can say that D x times X B is equal to it's equal to this thing right here it's the same thing as the transformation of X represented in coordinates with respect to be or in these non-standard coordinates so it's equal to the transformation of X represented in this coordinate system represented with recordin 'its with respect to B we see that right there but what is the transformation of X what is the transformation of X well that's the same thing as a times X right that's kind of the standard the standard transformation if we had standard if we X was represented in standard coordinates so this is equal to X and standard coordinates times the matrix a and then that will get us to this dot and standard coordinates but then we want to convert it to this non standard coordinates just like that now if we have this if we have this how can we just get 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 we have this this is the same thing as this and so if we apply if we apply or actually no we want to go the other way we have we have this we have that right there that's this right there and we want to get just this dot represented in regular standard coordinates so what do we do we multiply it by C so let me write it this way let me write if we multiply both sides of this equation times C what do we get 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 let me 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 C inverse we don't have an X here we have an ax here so C inverse times a X right the vector ax represented in this non standard coordinates is the same thing as multiplying the inverse of our change of basis matrix times the vector ax this will essentially if I have my vector a X 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 non standard basis now what is the vector X equal to what is the vector X equal to well the vector X is equal to our change of basis matrix times X represented in these non-standard 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 C times our non-standard coordinates for X just like that so what is the let me summarize it just because I want a little bit on this point right there it's going to add a little bit confused if I start off with the non-standard 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 non-standard transversus non-standard representation of the transformation of X the non-standard representation 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 non-standard coordinates now a times X and non-standard coordinates a times X and non-standard coordinates is the same thing as C inverse times a times X right if you think this is the same thing as this and so if you if you have this and you want to represent a non-standard coordinates you multiply it by C inverse so that you'll get that representation in non-standard coordinates and then finally we say look X X is the same thing as C times the non-standard coordinate representation of X so X we can replace we can replace X with that right there and so the big takeaway here 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 times the coordinates of X with respect to the basis B and just like that we have a version so D D must be equal to C inverse AC so the the if D is the transformation transformation matrix for T with respect with respect to the basis to the basis B to the basis B and and let me write here and C is the change the change of basis change of basis matrix for for B then and let's 1 we know a is the let me write that down and might as well this is our big takeaway and a is the transformation I'll write in shorthand transformation matrix matrix for T with respect with respect to the standard basis the standard basis then we can say this is the big takeaway that D our matrix D is equal to C inverse C inverse times a times C that's our big takeaway from this video which is really interesting I don't you 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 an arbitrary different basis that we can use to represent RN so we can have different Atrus 'is that represent the linear transformation under different coordinate systems and if we want to figure out those different Masek matrices for different coordinate systems we can essentially just construct construct the change of basis matrix for thee 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