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Current time:0:00Total duration:18:53

Video transcript

let's say we have some function f F and it's a mapping from the set X to Y so if I were to draw the set X right there that's my set X and then if I were to draw the set Y just like that we know and I've done this several videos ago that a function just associates any member of our set X so I have some member of my set X there if I apply the function to it or if we're dealing with vectors we could imagine instead of using the word function we would use the word transformation but it's the same thing we would associate with this element or this member of X a member of Y so that's why we call it a mapping it says hey that guy when I apply this function we're doing a different color this little member right there is associated this little member of X is associated with this member of Y and you would have if this is right here this is a capital X so let's say we call let me just let's say we call this a and let's call that B and we would say that the function where a is a member of X and B is a member of Y we would say that F of a is equal to B this is all a review of everything that we've learned already about functions now I'm going to define a couple of interesting functions the first one I guess it's really just one function I said it's a couple but I'll call it the identity function identity the identity function and this is a functional this is called a big capital I and it's identity function operates on some set so let's say this is the identity function on set X and it's a mapping from X to X and what's interesting about the identity function is that if you give it some a that is a member of X so let's say you give it that a the identity function applied to that member of X so the identity function of a is going to be equal to a so it literally just Maps things back to itself so the identity function if I were to draw it on this diagram right here would look like this it would look like when we pick a nice suitable color it would look like this we just kind of be a circle it just points back at the point that you start off with associates all points with themselves that's the identity function on X if a is pointed as it applies to the point a if you apply it to some other point in X we it would just refer back to itself now that's the identity function on X you could also have an identity function on Y so let's say that B is a member of Y so I do B right there then the Y identity function so this would be the identity function on Y applied to B we'll just refer back to itself it would just refer back to itself and so that would be equal to B this is the identity function on Y and so you might say hey Sal these are kind of silly functions but we'll use them they're actually a at least a useful notation to use as we kind of progress through our explorations of linear algebra but I'm going to make a new definition I'm going to say that a function a function let me pick a nice color pink I'm going to say that a function or let me say F since we already established it right over here I'm going to say that F is invertible invertible introducing some new terminology f is invertible if and only if the following is true so if and only if the following is true I could either write it with this two-way arrows like that or I could write it as if with two F's that means that if this is true then this is true and only if this is true and so this implies that and that implies this so f is invertible I'm kind of making a definition right here if and only if there exists there exists a function a function we'll call it will allow I'll call it nothing just yet I'll call it something in a second alright it is this F with this negative one superscript on it so f is invertible if and only if there exists a function f inverse well I guess I just called it something F inverse such that such that let me do that in purple such that if I apply F if I apply F remember F is just a mapping from X to Y so let me redo it so this function f inverse is going to be a mapping from Y to X so I'm saying that F is invertible if there exists a function f inverse that's a mapping from Y to X such that if I apply if I take the composition if I take the composition of F inverse with F this is equal to the identity the identity function over X so let's think about what's happening this is just part of it actually let me just complete the whole definition this is true this has to be true and F the composition of F with the identity function has to be equally the composition of F with the inverse function has to be equal to the identity function over Y so let's think about what's this saying there's some function f we'll just call it well I'll call it right now this is called the inverse of F this is the inverse the inverse of F and it's a mapping from Y to X so f was a mapping from let me draw it up here so f is a mapping from X to Y we showed that this is the mapping of F right there it goes in that direction we're saying there has to be some other function f inverse that's a mapping from Y to X so let's write it here so f inverse is a mapping from Y is a mapping from Y to X so f inverse if you give me some value and set Y I go to set X so this guy's domain is this guy is codomain and this guy's Co domain is this guy's domain fair enough but let's see what it's saying it's saying that the composition of F inverse with F has to be equal the identity matrix so essentially it's saying if I apply F to some value in X right if you think about what what's this composition doing this guy is going from X to Y and then this guy goes from Y to X right so let's think about what's happening here F F is going from X to Y and then F inverse is going from Y to X so this composition is going to be a mapping from X to X which the identity matrix or the identity identity function needs to do it needs to go from X to X and they're saying this equals the identity function so that means when you apply F on some value in our domain so you go here and then you apply f inverse to that point over there you go back to this original point so another way of saying this another way of saying this right here is that F let me do it in another color that F inverse the composition of F inverse with F of some member of X of the set X is equal to is equal to the identity function applied on that item this is what this these two statements are equivalent and so the ID by definition this thing is going to be your original thing or another way of writing this is that F inverse applied to F of a is going to be equal to a that's what this first statement tells us and if you think of it visually it's saying you start with an A you apply F to it and you get this value right here that is f of a I'm saying it equals B or I set it equal to B earlier on but then if you apply this F inverse and I doesn't always exist but if you apply that F inverse to this function it needs to go back to this by definition it needs to go back to original a it has to be equal has to be equivalent to just doing this little closed loop right here when I introduce you to the identity function now that's what this statement is telling us right here the second statement is saying look if I apply F to F inverse I'm getting the identity function on Y so if I start at some point in Y right there and I apply F inverse first if I apply F inverse maybe I go right here maybe this point let's call that lowercase Y so this would be F inverse of lowercase Y and then if I were to apply F to that I know this is getting this chart is getting very confusing if I apply F to this right here I need to go right back to my original Y so this when I apply F to F inverse of Y this has to be equivalent of just doing the identity function on Y so that's what the second statement is saying or another way to write it is that F of F inverse of let's say of Y where Y is a member of the set capital y it has to be equal to Y and you've been exposed the idea of an inverse before we're just doing it a little bit more precisely because we're where we're going to start dealing with these notions with transformations and matrices in the very near future so it's good to be exposed to it in this kind of more precise form now the first thing you might ask is hey let's say that I have a function f let's say I have a function f and it does have there does exist a function f inverse that satisfies these two these two requirements so f is invertible f is invertible the obvious question or maybe it's not an obvious question is is is f inverse unique and you actually probably the obvious question is how do you know when something is invertible and we're going to talk a lot about that in the very near future but let's say we know that f is invertible how do we know or do we know whether f inverse is unique and to answer that question to answer that question let's assume it's not unique so if it's not unique let's say that there's two functions that satisfy our two range that that can act as inverse functions of f so let's say that G is one of them so let's say G is a mapping remember F is a mapping from X to Y let's say that G is a mapping from Y to X such that if I apply F to something if I apply F to something and then apply G to it so this gets me from X to Y and then when I do the composition with G that gets me back into X this is equivalent this is equivalent to the identity function this was this was part of the definition of what it means to be an inverse so I'm saying that G is an inverse of I'm assuming that G is an inverse of F this assumption implies these two things that assumption applies those two things now let's say that H is another inverse let's say that H is another inverse of F H is another inverse by definition by what I just called an inverse H has to satisfy two requirements has to be a mapping from Y to X and then if I take the composition of H with F I have to get the identity matrix on the set X now that wasn't just part of the definition it implies even more than that if something is an inverse has to satisfy both of these the inverse the composition of the inverse with the function has to become the identity matrix on X and then the composition of the function with the inverse has to be identity matrix sorry the identity function on Y so let's write that so G is an inverse of F it applies this and it also implies it also implies I'll do it in yellow that the composition of F with G is equal to the identity matrix so I keep saying matrix the identity function on Y and if we do it with H the fact that H is an inverse of F implies that the composition of F with H is equal to the identity function unwise as well and just as Roman let me redraw what I drew at the beginning just so we know what we're doing so if this is the set X right here let me do it in different color let's say this right here is the set Y we know that F is a mapping from X to Y we know that F is a mapping from X to Y what we're seeing is what we're trying to determine is is f inverse unique so any inverse so we're saying that G is a situation that if you take the composition of G with F you get the identity matrix so f does that if you could take G if you take G you're going to go back to the same point so it's equivalent so taking the conversation of G with F that means doing F first then G this is equivalent this is equivalent of just taking the identity function in X so just taking an X and going back to an X it's equivalent to that so this is G right here and the same thing is true with H H should also be if I start with some element in X and go into Y and then apply H and then apply H like that it should also be equivalent to the identity transformation that's what this statement and this statement are saying now this statement is saying that if I start with some entry in Y here and I apply G which is the inverse of F I'm going to go here so G will take me there and that when I apply F then to that so then when I apply my F to that I'm going to go back to that same element of Y and that's equivalent to just doing the identity function on Y so that's the same thing as the identity function of Y and I could do the same thing here with H just take a point here apply H then apply F back I should just go back to that point that's all of what this is saying so let's go back to the question of whether G is unique are these can we have two different inverse functions G and H so let's start with G remember G is just a mapping from Y to X so this is going to be equal to this is the same thing as the composition of the identity function over X with G to show you why that's the case remember G just goes from let me all these diagrams get confused very quickly so let's say this is X and this is y remember G is a mapping from X to from Y to X so G will take us there is a mapping from Y to X and I'm saying that this G is equivalent to the identity mapping or the identity function in composition with this because all this is saying is you apply G and then you apply the identity mapping on X so obviously you're going to get to the exact same mapping or the exact same point so these are equivalent but what is another way of writing the identity mapping on X what's another way of writing that well by definition if H is another inverse of F this is true so I can replace this in this expression with a composition of H with F so this is going to be equal to the composition of H with F and the composition of that with G and you might want to put parentheses here actually I'll do it very lightly you might want to put parentheses there but I showed you a couple of videos ago that the composition of functions or of transformations is associative it doesn't matter if you put the parentheses there or if you put the parentheses there actually I'll do that I'll put the parentheses there at first just so you can understand that this this is the same thing as that right there but we know that composition is associated so this is equal to the composition of H with the composition of F and G now what is this equal to the composition of F and G well it's equal to by definition it's equal to the identity transformation over Y so this is equal to this is equal to H composed with of the composition of H with the identity function over Y with this right here now what is this going to be remember H is a function is a mapping from Y to X so let me redraw it so that's my X and that is my Y take some element in why and gives me some element in X now if I take the composition of well let me take it if I take the composition of the identity and Y so that's essentially I take some element let me do it this way I take some element in Y I apply the identity function which essentially just gives me that element again and then I apply H to that that's the same thing as just just applying H to the function to begin with to begin with so just going through this little exercise we've shown even though we've started off saying I have these two different inverses we've just shown that G must be equal to H so any function has a unique inverse you can't set up two different inverses and you'll if you do you'll find that they're always going to be equal to each other so so far we know what an inverse is and we know and we don't know what causes someone to be able to have an inverse or not but we know if they have an inverse how to think about it and we also know that that inverse we also know that that inverse is unique