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Current time:0:00Total duration:13:20

Alternate basis transformation matrix example

Video transcript

let's review a bit of what we learned in the last video if I have some linear transformation that's a mapping from RN to RN and if we're dealing with standard coordinates that transformation apply to some vector X in standard coordinates will be equal to the matrix a times X so let me write this down if we are dealing with the standard coordinates standard standard coordinates coordinates so I have X in standard coordinates if I apply the transformation that is equivalent to multiplying X by a then I if I multiply X by a then I'm going to get the transformation of X I'm going to get the transformation of X in standard coordinates this is a world that we're very very familiar with now let's say that we have an alternate basis to RN so let's say that B let's say B is equal to v1 v2 all the way to VN so it has n linearly independent vectors let's say B is a basis for RN so it's a basis for RN but it's the non standard basis these aren't just our standard basis vectors so B is a basis for RN and let's say let's say that C C which is just as these guys as its column vectors C is v1 v2 all the way to VN is the change of basis matrix change of basis matrix matrix for this basis for the basis B now we've learned we've seen this several times already that if I have if I have some vector X in RN represented in B coordinates or in coordinates with respect to B I can multiply it by the change of basis matrix and then I'll get just the standard coordinates for X or if you multiply both sides of this equation by C inverse you can get that if I start with the standard chord it's forex I can multiply it by C inverse and then I could get the B coordinates for X or the alternate non-standard coordinates for X so B but we've seen both of these before so let's apply that to this little diagram here so if I want to get X and if I wanted to write it in non-standard coordinates what do I do well if I have X and if I want to write it so if I have X let me go right here so if I have X what do I multiply it by if I want to go to non-standard coordinates well I multiply it by C inverse if I multiply it by C inverse you can whatever I write next to this line you can say what do you have to multiply what what matrix you have to multiply by to get to the of endpoint on your line so I multiply X by C inverse then I get then I get the V coordinates then I get the B coordinates for X so these are let's see coordinates coordinates with respect with respect to B I could do the same thing here with the transformation of X this is just the standard representation of the transformation of X so I could multiply it by C inverse if we want to go in that direction and then we're going to get the transformation of X we're going to get the transformation of X represented in B coordinates now in the last video what we saw is hey why go you know why do these separately maybe there is some matrix and we found out what it is maybe there's some matrix D and if we multiply this guy dot times it I can go straight from the B coordinates of X to the B coordinate of the transformation of X and we said that is matrix D and in that last video we show that D can be represented by a actually you could go around the circle and read rive it if you like but we found out that you write it in another color that D is equal to C inverse times a times C now this is all a review of everything that we learned in the last video hopefully clarify things up a little bit it's nice to just realize that these are just alternate ways of doing the same thing both of these are the transformation or regal form you multiply by a you're applying the same transformations when you multiply by D you're just doing it in a different coordinate system different coordinate systems are just different ways of representing the same vector this and this are same labels for the are different labels for the same vector this and this are different labels for the same vector so these are both performing the transformation T now this was a relation we got in the last video that if you give us if you if we have our change of basis matrix we have its inverse and we have just our standard basics linear trend our standard basis when your transformation matrix we were able to get this let's see if we can go the other way if we have D can we solve for a well if you multiply both sides of this equation on the right by C inverse you get D C inverse is equal to C inverse AC C inverse I just put a C inverse on the right hand side of both sides of this equation this is going to be the identity matrix so we can ignore it and then let's multiply both sides on the left by C so then you get C D C inverse is equal to C C inverse a and this is going to be the identity matrix and then you're left with a is equal to C times D C inverse which is another interesting result it's another thing to put in our toolkit now everything I've been doing has been fairly abstract let's actually apply some of these principles with a real concrete example so let's say that I have a transformation T let's say I have a transformation T I'll keep these guys around just because they might be useful that is a mapping from r2 to r2 and let's say that the transformation matrix for T so let's say that T of X in standard coordinates is equal to the matrix 3 2 minus 2 minus 2 minus 2 times X so this is in the example we just said this would be our transformation matrix with respect to the standard basis and we could call that a right there now let's say we have some alternate basis let's say we have some alternate basis so alternate alternate our two basis let's say let's call that B because we've been calling it B so far now let's say this alternate our two basis the vectors 1 2 & & 2 1 so let's see given this alternate basis whether we can come up for a transformation matrix in that coordinate world so what we're looking for we're looking for some matrix D we're looking for looking for a matrix D such that if I apply my transformation to X in B coordinates so if I apply it to X in B coordinates or in coordinates with respect to this alternate basis it should be equal to this matrix this matrix it should be equal to D times X X in the B coordinates so this is what I'm looking for I'm looking for that or if we go back to our diagram I'm looking for that you give me X and B coordinates and I and you multiply it by D and I'm going to give you the transformation of X and B coordinates now just applying it to this concrete example here all we we have this formula right here this is a formula for D which we proved in the last video so we have to figure out C inverse so what is the change of basis matrix for B we do it down I'm going to leave this up here so let me so change change of basis matrix matrix for B is just going to be let's go call it C and it's going to be the basis vectors for B's within the column so 1 2 and 2 1 and then we're going to want to figure out its inverse so let's figure out its determinant first so the determinant of C is equal to 1 times 1 minus 2 times 2 so 1 minus 4 is minus 3 and so C inverse is going to be equal to 1 over the determinant 1 over minus 3 or minus 1 thirds times we switch these two guys so we switch the 1 on the 1 and then we make these two guys negative minus 2 minus 2 that is C inverse so D this D get this D vector right here is going to be equal to C inverse times a time's the transformation matrix with respect to the standard basis times C so let me write it down here just because let me write it so D the D that we're looking for D is going to be equal to C inverse times a times C which is equal to C inverse is minus 1/3 times 1 minus 2 minus 2 1 times times a let me do this in a different color I like to switch colors so C inverse times a a is right there so times 3 minus 2 2 minus 2 times C C is right there I'll do it in yellow times C which is 1 2 and then 2 1 so let's do this piece by piece so work through this so what is this piece going to be equal to we have a 2 by 2 times the 2 by 2 that's going to give us another 2 by 2 matrix so this first term right here is will be 3 times 1 plus minus 2 times 2 so 3 3 minus 4 so it's going to be minus 1 right 3 times 1 plus minus 2 times 2 right it's minus 1 then you have 3 times 2 which is 6 minus 2 right minus 2 times 1 so that is 4 right 3 times 2 minus 2 is 4 and then when you go down here 2 times 1 minus 2 times 2 that's 2 minus 4 that's minus 2 and then 2 times 2 is 4 minus 2 times 1 4 minus 2 is just 2 so our matrix D is going to be equal to minus 1/3 times this guy 1 minus 2 minus 2 1 times this guy which was just the product of those two matrices now let's figure out what this is if I take the product of these two guys it's going to be another 2x2 matrix so if 1 times minus 1 1 times minus 1 which is minus 1 plus minus 2 times minus 2 minus 2 times minus 2 so let me make sure so 1 times minus 1 minus 2 times minus 2 is 4 and then 1 times minus 1 is minus 1 so it's going to be 3 and then we have and then we go to the next term we have 1 times 4 plus minus 2 times 2 so that's 4 minus 4 which is 0 and then we have minus 2 times minus 1 which is 2 plus 1 times minus 2 so that is 0 and then finally we have minus 2 times 4 which is minus 8 right minus 2 times 4 is minus 8 plus 1 times 2 so minus 8 minus 2 times 4 is minus 8 plus 2 is minus 6 and all of that times minus 1/3 so this is going to be equal to 3 times minus 1/3 is minus 1 0 and then 0 minus 6 times minus 1/3 is 2 is 2 so this is so d is now our transformation matrix is now our transformation matrix with respect to the basis B with respect to the basis B so we were able to figure it out just applying this formula here now what happens let's actually do it with some actually I'll save that for the next video where we actually show that it works that it we can actually take some vectors we can actually take some vectors X apply the transformation or apply the change of coordinates to get to this and then apply D and then maybe we could go up that way multiply by C to get the transformation it's going to be equivalent to a I'll do that in the next video