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# Showing that inverses are linear

## Video transcript

I've got a transformation T and when you apply T the transformation T to some X in your domain it is equivalent to multiplying that X in your domain or that vector by the matrix a now let's say we know that the linear transformation T can be that's transformation matrix when you put it in reduced row echelon form the transformation matrix it is equal to an N by n identity matrix or the N by n identity matrix well this alone this alone tells us a lot of things first of all if when you put this in reduced row echelon form you get a square identity matrix that tells us that the original matrix had to be n by n it also tells us that T is a mapping from RN to RN and we saw in the last video all of these are conditions especially this one especially this one right here for T to be invertible for t to be invertible so if we know that this is true T is a linear transformation it's the reduced row echelon form of its transformation matrix is the identity matrix right there we know that T is invertible invertible but let's remind ourselves what it even means to be invertible to be invertible means means that there exists there exists some we use the word function before but now we're talking about transformations they're really the same thing so let's say there exists some transformation some transformation let's call it T inverse T inverse like that or T to the minus 1 such that such such that the composition of t-inverse t-inverse with t is equal to the identity transformation on your domain the identity transformation on your domain and the composition of T with T inverse is equal to the identity transformation on your codomain on your co-domain just like that and just to remind you what this looks like let's draw our domains and domain our domain is our n and our codomain is also our n so it's just like that Co domain is also our n so if you take some if you take some vector in your domain apply the transformation T you're going to go into your Co domain so that is T and then if you apply that the T inverse after that you're going to go back to that original X so this is this look you apply T and then you apply T inverse you're just going to get back to where you start it's equivalent to the identity transformation it's equivalent to the identity transformation just like that this is saying if you start here and your codomain you apply your inverse first you apply this inverse transformation first then you apply your transformation you're going to go back to the same point in your codomain so it's equivalent to the identity transformation in your codomain it just happens to be in this case that the domain and the co domain are the same set are and now we know what a transformation what it means to be invertible we know whether the conditions are for invertibility so this begs to the next question is this we know that this guy's is a linear transformation in fact that's one of the conditions to be able to represent it as a matrix or any transformation that can be represented as a matrix vector product is a linear transformation so this guy is a linear transformation but the question is is is T inverse a linear transformation a linear transformation now let's review what the two conditions are that we need to have to be a linear transformation so we know T is a linear transformation so we know that if you apply the transformation T to two vectors let's say x and y if we apply to the sum of those two vectors it is equal to the transformation of the first vector plus the transformation of the second vector that's one of the conditions or one thing that we know is true for all linear transformations and the second thing we know for all linear that we know is true for all linear transformations is if we take the transformation of some scaled version of a vector in our domain it is equal to the scaling factor times the transformation of the vector itself these are both conditions for linear transformations so let's see if we can prove that both of these conditions hold for T inverse for this guy right here so to do this let's just let's supply let's do this little exercise right here let's apply T let's take the composition of t with t-inverse of of two vectors a plus B remember T inverse is a mapping from your from your co-domain to your domain although they're both going to be RN in this case but T inverse maps from this set to that set let's write it up here T inverse is a mapping from your codomain to your domain although it looks identical just like that okay so what is this going to be equal to well we just said by definition your inverse transfer of your inverse transformation this is going to be equal to the identity transformation on your codomain so assuming that these guys are members of your codomain in this case RN this is just going to be equal to this is just going to be equal to a plus B this thing the composition of T with its inverse by definition is just the identity transformation on your codomain so this is just whatever I put in here if I put in an X here this would be an X if I put in an apple here this would be an apple it's going to be the identity transformation now what is this equal to what is this equal to well I could use the same argument to say that this right here is is equal to the identity transformation applied to a and I'm not writing the identity transformation I'm writing this but we know that this is equivalent to the identity transformation so we could say that equivalent to the composition of T with the inverse applied to a and we could say that this is the equivalent to the identity transformation which we know is the same thing as T the composition of t with t-inverse applied to B so we can rewrite this thing right here we can rewrite this thing right here as being equal to these two theta some of these two things in fact we don't even have to rewrite it we can just write it's equal to this transformation is equal to this and maybe an easier way for you to to to I guess process it is we could write this as T of the T inverse of a plus B is equal to T of the T inverse of a plus T of the T inverse of B and this should you know that I don't know which one your brain processes easier but either of these when you take the competition of t with t-inverse you're just going to be left with an A plus B you take the composition of t with t-inverse you're left with an A you take the composition of t with t-inverse you're just left with a be there so in either case you get a plus B the vector A plus you get when you evaluate either side of this expression you'll get the vector A plus the vector B now what can we do now what can we do we know that T itself is a linear transformation T is a linear transformation and since T is a linear transformation we know that T applied to the sum of two vectors T applied to the sum of two vectors is equal to the sum of is equal to T applied to each of those vectors and summed up or we could take it the other way T applied to two separate vectors so that we could call this one vector right here and this vector right here so in this case I have a T applied to one vector and I'm summing it to a T applied to another vector so it's this right here which we know is equal to T applied to the sum of those two vectors so this is T applied to the vector T inverse of a let me write it here so this is going to be equal to T inverse of a plus T inverse of B it might look a little convoluted but all I'm saying is look this looks just like this if you say that if you say that X is equal to T inverse of a and if you say that Y is equal to T inverse of B so this looks just like that it's going to be equal to the transformation T applied to the sum of those two vectors so it's going to equal to the transformation T applied to the inverse of a T inverse of a plus T inverse of B I just use the fact that T is linear to get here now what can I do let me let me simplify everything that I've written right here so I now have let me rewrite this this thing up here which is the same thing as just T the composition of t with t-inverse applied to a plus b applied to a plus b is equal to the composition or not and actually not the composition just T apply to two vectors T inverse of A plus T inverse of vector B that's what we've gotten so far now we're very close to proving that this condition is true for T inverse if we can just get rid of these T's well the best way to get rid of those T's is apply to T is to take the composition with T inverse on both sides or take the T inverse transformation of both sides of this equation so let's do that so let's take T inverse of this side T inverse of that side should be equal to T inverse of this side because these two things are the same thing so if you put the same thing into a function you should get the same value on both sides so what is this thing on the left-hand side what is this this is the composition right this way this is the composition of t-inverse with t that part applied to this thing right here apply to I'm just changing the associativity of this apply to T inverse of the vector A plus the vector B that's what this left-hand side is this part right here T inverse of T of this on just this first two steps I'm just writing as a composition of t-inverse with t applied to applied to this right here that right there is the same thing as that right there so that was another way to write that and so that is going to be equal to that is going to be equal to the composition of t-inverse with t I'll write it in the same color of t-inverse with t that's this part right here which is very similar to that part right there of of this stuff right here of t-inverse of A plus T inverse of the vector B now by definition of what T inverse is what is this this is this is the identity transformation on our domain this is the identity transformation on RN this is also the identity transformation on RN so if you apply the identity transformation to anything you're just going to get anything so this is going to be equal to I'll do it on both sides of the equation this is just going to this whole expression on the left-hand side is just going to simplify to the T inverse of the vectors a plus the vector B and the right-hand side is just going to simplify to this thing is equal to because this is just the identity transformation so it's just equal to this one P inverse of the vector a plus T inverse of the vector B and just like that we've met T inverse has met its first condition for being a linear transformation its met its first condition now let's see if we can do the second condition let's take let's take T of let's do the same type of thing let's take the composition of t with t-inverse of let's take the composition of that on some vector let's call it C a just like that well we know what this is equal to this is equal to the identity transformation on RN so this is just going to be equal to C a now what is a equal to what is a equal to what is this thing right there equal I'll write it on the side right here let me do it in a 4-bit color we could say that a the vector a is equal to the transformation T with the composition of T with T inverse T inverse applied to the vector a because this is just the identity transformation so we can rewrite this expression here as being equal to C times the composition of the composition of t with t-inverse applied to apply to my vector a and maybe it might be nice to rewrite it in this in this form instead of this composition form so this left expression we can just write it saying T of the T inverse of C times the vector a all I did is rewrite this left-hand side this way is equal to is equal to this green thing right here well I'll rewrite similarly this is equal to C times T the transformation T applied to the transformation T inverse applied to a this is by definition what composition means this is what composition means now T is a linear transformation T is a linear transformation which means that you can take that if you take C times T times some vector that is equivalent to T times C times T applied to C times that vector this is one of the conditions of a linear transformation so this is always going to be the case with T so this is some vector that T is applying to this is some scalar so this thing because we know that T is a linear transformation we can rewrite as being equal to T applied to the scalar C times T inverse apply to a applied to a and now what can we do well let's apply the T inverse transformation to both sides of this let me rewrite it so on this side we get T of T inverse of C a is equal to T of C times T inverse times a that's what we have so far and wouldn't it be nice if we could get rid of these outer T's and the best way to do that is to take the inverse the T inverse transformation of both sides so let's do that t inverse let's take that of both sides of this equation T inverse of both sides and another way that this could be written this is equivalent to T the composition of t-inverse with t applied to t-inverse applied to C times our vector a this right here I've just decided to keep writing it in this form and I took these two guys out and I wrote them as a composition and this on the right-hand side you can do something very similar you could say that this is equal to the composition of t-inverse with t times or not times let me be very grateful taking this composition this transformation and then taking that transformation on on C times C times the Trant the inverse transformation applied to a let me very clear what I did here this thing right here is this thing right here this thing right here is this thing right here and I just rewrote this composition this way I rewrote this composition this way and the reason why I did this is because because we know that this is just the identity transformation on RN and this is just the identity transformation on RN so the identity transformation applied to anything is just that anything so this equation simplifies to the in T inverse applied to C times some vector a is equal to this thing C times T inverse times some vector a and just like that we've met our second condition or being a linear transformation we've met our second condition the first condition was met up here so now we know and in both cases we use the fact that T was a linear transformation to get to the result for T inverse so now we know that if T if T is a linear transformation T is a linear transformation and T is invertible invertible then T inverse is also also a linear transformation linear transformation which might seem like a little nice thing to know but that's actually a big thing to know because now we know that T inverse can be represented as a matrix vector product so that means that T inverse applied to some vector X could be represented as the product of some matrix times X and what we're going to do is we're going to call that net matrix we're going to call that matrix the matrix a inverse and I haven't defined this well how do you construct this a inverse matrix but we know that it exists we notice this exists now because T is the linear transformation and we could take it even a step further we know by the definition of a of invertibility that the composition of t-inverse with t is equal to the identity transformation on r and the identity transformation on r and well what is the composition we know that T if we take if we take let me put it this way we know that T of X is equal to ax so if we write T inverse the composition of t-inverse with t applied to some vector X is going to be equal to first ax a being applied to X so it's going to be equal to ax this a right here ax and then you're going to apply a inverse X you're going to apply this right here and we got this that this is equivalent to when you take the composition it's equivalent to or the resulting the resulting transformation matrix of two composition transformations is equal to this matrix matrix product we got that a long time ago in fact that was the motivation for how a matrix matrix product was defined but what's interesting here is this composition is equal to that but it's also equal to it's also going to be equal to the identity transformation on RN applied to that vector X which is equal to the identity matrix entity matrix applied to X right that is the N by n matrix that when you multiply it by anything you get that anything again so we get a very interesting result a inverse times a has to be equal to the identity matrix a inverse or the matrix transformation for T inverse when you multiply that with the matrix transformation for T you are going to get the identity matrix and the argument actually holds both ways so we know this is true but the other definition of an inverse or invertibility told us that the composition of T with T inverse is equal to the identity transformation in our codomain which is also RN ir n so by the exact same argument we know that when you go the other way if you apply T inverse first and then you apply T so that's the equivalent of saying you apply T inverse first and then you apply T to some X vector that's equivalent to multiplying that X vector by the identity matrix the N by n identity matrix or you could say you could switch the order a times a inverse is also equal to the identity matrix which is neat because we learned that matrix matrix products when you switch the order they don't normally always equal each other but in the case of a of an invertible matrix and its inverse order doesn't matter you can take a inverse times a and get the identity matrix or you could take a times a inverse and get any matrix now we've gotten this far the next step is to actually figure out how do you construct we know that this thing exists we know that the inverse is linear transformation that this matrix exists we see this nice property that when you multiply it times the transformation matrix you get the you get the identity matrix the next step is to actually figure out how to figure this this guy out