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So what that tells us, just to make sure we are interpreting our eigenvalues and eigenspaces correctly, is look, you give me any eigenvector in this. You give me any vector right here. Let's say that is vector x. If I apply the transformation, if I multiply it by a, I'm going to have 3 times that, because it's in the e... | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
If I apply the transformation, if I multiply it by a, I'm going to have 3 times that, because it's in the eigenspace for lambda is equal to 3. So if I were to apply a times x, a times x would be just 3 times that. So that would be a times x. That's what it tells me. This would be true for any of these guys. If this was... | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
That's what it tells me. This would be true for any of these guys. If this was x, and you took a times x, it's going to be 3 times as long. Now these guys over here, if you have some vector in this eigenspace that corresponds to lambda is equal to 3, and you apply the transformation, let's say that this is x right ther... | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Now these guys over here, if you have some vector in this eigenspace that corresponds to lambda is equal to 3, and you apply the transformation, let's say that this is x right there. If you took the transformation of x, it's going to make it 3 times longer and in the opposite direction. It's still going to be on this l... | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
That's just some set or some subset of Rn where if I take any two members of that subset, so let's say I take the members a and b, they're both members of my subspace. By the fact that this is a subspace, we then know that the addition of these two vectors, or a plus b, is also in my subspace, and this is our closure u... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
And we sometimes call this closure under scalar multiplication. And then a somewhat redundant statement is that V must contain the zero vector. And that's true of all subspaces. V, let me write it this way, the zero vector is a member of V, and it would be the zero vector with n components here, because V is a subspace... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
V, let me write it this way, the zero vector is a member of V, and it would be the zero vector with n components here, because V is a subspace of Rn. And why I say that's redundant, because if I say that any multiple of these vectors is also in V, I could just set the scalar to be equal to 0. So this statement kind of ... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
Fair enough. Now, let's say that I also have some transformation T. It is a mapping, a function from Rn to Rm. What I want to understand in this video is I have a subspace right here, V. I want to understand whether the transformation of the subspace, and what did we call that, we call that the image of our subspace or... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
It was a triangle that looked something like that, and we figured out its image under T. So we went from R2 to R2, and we had our transformation, and it ended up looking something like this, if I remember it properly. Actually, I don't remember it fully, but it was like a triangle that was skewed like this, rotated. So... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
But the exact particulars of that last video aren't what matter. What matters is that you are able to visualize what an image under transformation means. It means you take some subset of R2, all of the vectors that define this triangle right here, that some subset of R2. You transform all of them, and then you get some... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
You transform all of them, and then you get some subset in your co-domain. And this is the, you could call this the image, call this the transformation of that triangle, or if we call this S, it's equal to the transformation of S. Or you could say it's the image of, you could call it the set S, but maybe it helps you t... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
And just as you could take scalar multiples of some of the vectors that are subsets, that are members of this triangle, and you'll find that they're not going to be in that triangle. So this wasn't a subspace. This was just a subset of R2. All subsets are not subspaces, but all subspaces are definitely subsets, althoug... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
All subsets are not subspaces, but all subspaces are definitely subsets, although something can be a subset of itself. I don't want to wander off too much. But this just helps you visualize what we mean by an image. It means all of the vectors that are mapped to from the members of your subset. So I want to know whethe... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
It means all of the vectors that are mapped to from the members of your subset. So I want to know whether the image of V under T is a subspace. So in order for it to be a subspace, if I take the transformation, let me find two members of T. Well, clearly, if I take the transformation of any members of V, I'm getting me... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
So I can write this. Clearly, the transformation of A and the transformations of B, these are both members of our images of V under T. These are both members of that right there. So my question to you is, what is the transformation of A plus the transformation of B? And the way I've written this, these are two arbitrar... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
And the way I've written this, these are two arbitrary members of our image of V under T. Or maybe I should call it T of capital V. These are two arbitrary members. So what is this equal to? Well, we know from our properties, our definition of linear transformations, this is equal to the sum of the transformations of t... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
Now, is the transformation of A plus B a member of TV? Is it a member of our image? Well, A plus B is a member of V, and the image contains the transformation of all of the members of V. So the image contains the transformation of this guy. This guy, A plus B, is a member of V. So you're taking the transformation of a ... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
This guy, A plus B, is a member of V. So you're taking the transformation of a member of V, which by definition is in your image of V under T. So this is definitely true. Now, let's ask the next question. If I take a scalar multiple of some member of my image of V under T, or my T of capital V right there, what is this... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
By definition for linear transformation, this is the same thing as a transformation of the scalar times the vector. Now, is this going to be a member of our image of V under T? Well, we know that CA is definitely in V. That's from the definition of a subspace. This is definitely in V. And so if this is in V, the transf... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
This is definitely in V. And so if this is in V, the transformation of this has to be in V's image under T. So this is also a member of V. And obviously, you can set this equal to 0. The 0 vector is a member of V, so any transformation of, if you just put a 0 here, you'll get the 0 vector. So the 0 vector is definitely... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
So the 0 vector is definitely also a member of TV. So we come on the result that the image of V under T is a subspace, which is a useful result, which we'll be able to use later on. But this, I guess, might naturally lead to the question, what if we go, everything we've been dealing with so far have been subsets, in th... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
This is the image of Rn under T. Let's think about what this means. This means, what do we get when we take any member of Rn, what is the set of all of the vectors, when we take the transformation of all of the members of Rn? Let me write this. This is equal to the set of the transformation of all of the x's where each... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
This is equal to the set of the transformation of all of the x's where each x is a member of Rn. So you take each of the members of Rn and transform them, and you create this new set. This is the image of Rn under T. Well, there's a couple of ways you can think of this. Remember when we defined, let's see, T is a mappi... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
Remember when we defined, let's see, T is a mapping from Rn to Rm. We defined this as the domain, all of the possible inputs for our transformation, and we defined this the codomain. Remember I told you that the codomain is essentially part of the definition of the function or of the transformation, and it's the space ... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
It's not necessarily all of the things that we're mapping to. For example, the image of Rn under transformation, maybe it's all of Rm, or maybe it's some subset of Rn. The way you can think about it, and I touched on this in that first video, and at least the linear algebra books I looked at, they didn't specify this, ... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
That if you take the image of Rn under T, you're actually finding, so let's say that Rm looks like that. Obviously it'll go in every direction. And let's say that when you take, let me draw Rn right here, say this is Rn, and we know that T is a mapping from Rn to Rm, but let's say when you take every element of Rn and ... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
So let me see if I can draw this nicely. So you literally map every point here, and it goes to one of these guys, or one of these guys can be represented as a mapping from one of these members right here. So if you map all of them, you get this subset right here. This subset is T, the image of Rn under T, and in the te... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
This subset is T, the image of Rn under T, and in the terminology that you don't normally see in linear algebra a lot, you can also kind of consider it its range. The range of T. Now, this has a special name. This is called, and I don't want you to get confused, this is called the image of T. It might be a little confu... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
You're like, before when we were talking about subsets, we would call this the image of our subset under T, and that is the correct terminology when you're dealing with a subset. But when you take all of a sudden the entire n-dimensional space, and you're finding that image, we call that the image of the actual transfo... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
Well, we know that we can write any, and this is literally any. So T is going from Rn to Rm. We can write T of x, we can write any linear transformation like this as being equal to some matrix, some m by n matrix, m by n, times a vector. And these vectors obviously are going to be members of Rn. Rn. Times some Rn. And ... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
And these vectors obviously are going to be members of Rn. Rn. Times some Rn. And what is this? So what is the image? Let me write it in a bunch of different ways. What is the image of Rn under T? | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
And what is this? So what is the image? Let me write it in a bunch of different ways. What is the image of Rn under T? So we could write that as T of Rn, which is the same thing as the image of T. Notice we're not saying under anything else, because now we're saying the image of the actual transformation, which we coul... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
What is the image of Rn under T? So we could write that as T of Rn, which is the same thing as the image of T. Notice we're not saying under anything else, because now we're saying the image of the actual transformation, which we could also write as the image of T. Well, what are these equal to? These are equal to the ... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
Well, all the transformations of x are going to be Ax, where x is a member of Rn. So x is going to be an n-tuple where each element just has to be a real number. So what is this? So if we write A, let me write my matrix A, it's just a bunch of column vectors, A1, A2, it's going to have n of these, because it has n colu... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
So if we write A, let me write my matrix A, it's just a bunch of column vectors, A1, A2, it's going to have n of these, because it has n columns. And so A times any x is going to be, so if I multiply that times any x that's a member of Rn, so I multiply x1, x2, all the way to xn, we've seen this multiple, multiple time... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
And we're saying we want the set of all of these sums of these column vectors where x can take on any vector in Rn, which means that the elements of x can take on any real scalar values. So the set of all of these is essentially all of the linear combinations of the columns of A, right? Because I can set these guys to ... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
So what is that equal to? That is equal to, and we touched on this, or we actually talked about this when we introduced the idea, this is equal to the column span, sorry, the column space of A, or we just denoted it sometimes as C of A. So that's a pretty neat result. If you take, well, it's almost obvious, I mean, it'... | im(T) Image of a transformation Matrix transformations Linear Algebra Khan Academy.mp3 |
We saw this several videos ago. And the definition of our linear transformation, or the composition of our linear transformation, so the composition of s with t applied to some vector x in our set x, our domain, is equal to s of t of x. This was our definition. And then we went on and we said, look, if s of x can be re... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
And then we went on and we said, look, if s of x can be represented as the matrix multiplication A x, the matrix vector product, and if t of x can be represented, or the transformation t can be represented as the product of the matrix B with x, we saw that this thing right here, which is, of course, if we just write it... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
So you take, essentially, the first linear transformation in your composition, its matrix, which was A, and then you take the product with the second one. Fair enough. All of this is review so far. So let's take three linear transformations. Let's say that I have the linear transformation H, and when I apply that to a ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
So let's take three linear transformations. Let's say that I have the linear transformation H, and when I apply that to a vector x, it's equivalent to multiplying my vector x by the matrix A. Let's say I have the linear transformation G. When I apply that to a vector x, it's equivalent to multiplying that vector. There... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
There should be a new concept called a vectrix. It's equivalent to multiplying that vector times the matrix B, and then I have a final linear transformation F. Applied, when it's applied to some vector x, it's equivalent to multiplying that vector x times the matrix C. Now, what I'm curious about is what happens when I... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
So let's explore what this is a little bit. Well, by the definition of what composition even means, we can just apply that to this right here. So we could just imagine this as being our S. If we imagine this was our S, and then this is our T right there, then what is this going to be equal to? If we just do a straight ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
If we just do a straight up pattern match right there, this is going to be equal to S, the transformation S, applied to the transformation F applied to x. So S is H of G. So it is H, or I shouldn't say H of G, the composition of H with G, that is our S. And then I apply that to F applied to x. F is our T. I apply that ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
Now we can imagine that this is our x, if we just pattern match according to this definition, that this is this guy right here, that this is our T, and that this is our S. And so if we just pattern match here, this is equal to what? This is just straight from the definition of a composition. So it's equal to S of, S is... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
But instead of an x, we have a whole, this vector here, which was the transformation F applied to x. So G of F of x. That's what this is equal to. The composition of H with G, or the composition of F with the composition of H and G, all of that applied to x is equal to H of G of F of x. Now what is this equal to? What ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
The composition of H with G, or the composition of F with the composition of H and G, all of that applied to x is equal to H of G of F of x. Now what is this equal to? What is this equal to? Well, this is equal to, I'll do it right here, this is equal to H, the transformation H applied to, what is this term right here?... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
Well, this is equal to, I'll do it right here, this is equal to H, the transformation H applied to, what is this term right here? I'll do it in pink. What is this? That is the composition of G and F applied to x. You can just replace S with G and F with T and you'll get that right there. So this is just equal to the co... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
That is the composition of G and F applied to x. You can just replace S with G and F with T and you'll get that right there. So this is just equal to the composition of G with F applied to x. That's all that is. Now, what is this equal to right there? And it's probably confusing to see two parentheses in different colo... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
That's all that is. Now, what is this equal to right there? And it's probably confusing to see two parentheses in different colors, but I think you get the idea. What is this equal to? Well, just go back to your definition of the composition. I just want to make it very clear what we're doing. This is, if you imagine t... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
What is this equal to? Well, just go back to your definition of the composition. I just want to make it very clear what we're doing. This is, if you imagine this being your T and then this being your S, this is just the composition of S with T applied to x. So this is just equal to, I'll write it like this way, this is... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
This is, if you imagine this being your T and then this being your S, this is just the composition of S with T applied to x. So this is just equal to, I'll write it like this way, this is equal to, I just shouldn't write S's, this is the composition of H with the composition of G and F. And then all of that applied to ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
Well, one, to show you that the composition is associative. I went all the way here and then I went all the way back. And essentially, it doesn't matter where you put the parentheses. The composition of the composition of H with G with F is equivalent to the composition of H with the composition of G and F. These two t... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
The composition of the composition of H with G with F is equivalent to the composition of H with the composition of G and F. These two things are equivalent. And essentially, you can just rewrite them. The parentheses are essentially unnecessary. You can write this as the composition of H with G with F, all of that app... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
You can write this as the composition of H with G with F, all of that applied to x. Now, I took the time to say that I can represent H as a matrix, that each of these linear transformations I can represent as matrix multiplications. Why did I do that? Well, we saw before that any composition, when you take the composit... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
Well, we saw before that any composition, when you take the composition of S with T, the matrix version of this transformation, of this composition, is going to be equal to the product, by our definition of matrix-matrix products, the product of the S's transformation matrix and T's transformation matrix. So what are t... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
So let me write that, the matrix version of the composition of H with G, and then the composition of that with F applied to x, is going to be equal to, and we've seen this before, the product of these matrices. So this composition, its matrix is going to be AB. H and G, their matrices are A and B. So it's going to be A... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
So it's going to be AB and I'll do it in parentheses. And then you take that matrix, and you take the product, so this guy's matrix representation is AB, and this guy's matrix representation is C. So the matrix representation of this whole thing is this guy taking the product of AB and then taking the product of that w... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
That's the vector x. Now let's look at this one right here. If we take the composition of H with the composition of G and F and apply all of that to some vector x, what is that equivalent to? Well, this composition right here, the matrix version of it I guess we can say, is going to be the product BC. And we're going t... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
Well, this composition right here, the matrix version of it I guess we can say, is going to be the product BC. And we're going to apply that to x. So we're going to have the product BC. And then we're going to take the product of that with this guy's matrix representation, which is A. And we've shown this before. We ne... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
And then we're going to take the product of that with this guy's matrix representation, which is A. And we've shown this before. We never showed it with 3, but it extends. I mean, I kind of showed it extends. You can just keep applying the definition. You can keep applying this property right here, and so it'll just na... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
I mean, I kind of showed it extends. You can just keep applying the definition. You can keep applying this property right here, and so it'll just naturally extend. Because every time we're just taking the composition of 2 things, even though it looks like we're taking the composition of 3, we're taking the composition ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
Because every time we're just taking the composition of 2 things, even though it looks like we're taking the composition of 3, we're taking the composition of 2 things first here, and then we get its matrix representation. And then we take the composition of that with this other thing. So the matrix representation of t... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
Similarly, here we take first the matrix, the composition of these 2 linear transformations, and their matrix representation will be that right there. And then we take the linear, we take the composition of that with that. So the entire matrix representation is going to be this guy's matrix times this guy's matrix. So ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
So A times BC. And of course, all of that applied to the vector x. Now, in this video, I've showed you that these 2 things are equivalent. If anything, the parentheses are completely unnecessary. And I showed you that there. They both essentially come boiled down to h of g of f of x, so these 2 things are equivalent. S... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
If anything, the parentheses are completely unnecessary. And I showed you that there. They both essentially come boiled down to h of g of f of x, so these 2 things are equivalent. So we could say, essentially, that these 2 things over here are equivalent. Or that AB, the product AB, and then taking the product of that ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
So we could say, essentially, that these 2 things over here are equivalent. Or that AB, the product AB, and then taking the product of that matrix with the matrix C, is equivalent to taking the product A with the matrix BC, which is just another product matrix. Or another way of saying it is that these parentheses don'... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
Or, I mean, this is just a statement, that matrix products exhibit the associative property. It doesn't matter where you put the parentheses. And sometimes it's confusing the word associative. It just means it doesn't matter where you put the parentheses. Matrix products do not exhibit the commutative property. We saw ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
It just means it doesn't matter where you put the parentheses. Matrix products do not exhibit the commutative property. We saw that in the last video. We cannot, in general, make the statement that AB is equal to BA. We cannot do that. And in fact, in the last video, I think it was the last video, I showed you that if ... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
We cannot, in general, make the statement that AB is equal to BA. We cannot do that. And in fact, in the last video, I think it was the last video, I showed you that if AB is defined, sometimes BA is not even defined. Or if BA is defined, sometimes AB isn't defined. So it's not commutative. It is associative, though. I... | Matrix product associativity Matrix transformations Linear Algebra Khan Academy.mp3 |
And then my set that I'm mapping into, set Y, that's the co-domain. We know that T is a transformation that if you take any member of X and you transform it, you'll get, or you'll associate it with a member of set Y. You'll map it to a member of set Y. That's what the transformation or the function does. Now, if we hav... | Preimage of a set Matrix transformations Linear Algebra Khan Academy.mp3 |
That's what the transformation or the function does. Now, if we have some subset of T, let's call A to be some subset of T. So let me draw A like that. This notation right here just means subset. Some subset of T. We've defined the notion of an image of T, of A like that, which is the image of A, of our subset A under ... | Preimage of a set Matrix transformations Linear Algebra Khan Academy.mp3 |
Some subset of T. We've defined the notion of an image of T, of A like that, which is the image of A, of our subset A under T. We've defined this to be equal to the set, let me write it here, the set of all, where if we take each of the members of our subset, so if we take each of the members of our subset, it's the se... | Preimage of a set Matrix transformations Linear Algebra Khan Academy.mp3 |
You take another member of A, this is all set A right here, take another member of A, find its transformation, maybe it's that point. You keep doing that, find its transformation, maybe it's that point. And then the set of all of those transformations, maybe it's this blob right here, we call this the image of A under ... | Preimage of a set Matrix transformations Linear Algebra Khan Academy.mp3 |
What if we were to start with set Y, which is our codomain, so that's Y, and we were to have some subset of Y, let's call our subset of Y S. So S is a subset of our codomain Y, and I'm curious about what subset of X maps into S. I'm curious about this set, I'm curious about the set of all vectors that are members of my... | Preimage of a set Matrix transformations Linear Algebra Khan Academy.mp3 |
I'm not saying that every point in S necessarily gets mapped to. For example, maybe there's some element in S right here, right there, that no element in X ever gets mapped to from our transformation T. That's okay. All I'm saying is, is that everything in this set maps to something within S right here. And what we cal... | Preimage of a set Matrix transformations Linear Algebra Khan Academy.mp3 |
And what we call this, we call this set right here, we call this, the notation is the inverse T of S, but this is equal to the preimage of S under T. So this is S, this is the preimage of S under T, and that makes sense. The image, we go from a subset of our domain to a subset of our codomain. Preimage, we go from a su... | Preimage of a set Matrix transformations Linear Algebra Khan Academy.mp3 |
Now, let me ask you an interesting question, and this is kind of for bonus points. What is the image of our preimage under S? So if we take this guy, this is essentially the image of this guy right here. This part right here is the preimage of S right here. Now if we take the image of this, we're saying what, if we tak... | Preimage of a set Matrix transformations Linear Algebra Khan Academy.mp3 |
Let's say we have some function 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... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
So that's why we call it a mapping. It says, hey, that guy, when I apply this function, let me do it in a different color, this little member right there is associated, this little member of x is associated with this member of y. And if this is right here, this is a capital X. So let's say we call this a, and let's cal... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
So 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 fi... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
Now I'm going to define a couple of interesting functions. The first one, and I guess it's really just one function, I said it's a couple, but I'll call it the identity function. And this is a function, I'll just call it a big capital I, and its identity function operates on some set. So let's say this is the identity ... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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 e... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
So the identity function, if I were to draw it on this diagram right here, would look like this. It would look like, let me pick a nice suitable color, it would look like this. It would just kind of be a circle. It just points back at the point that you started off with, it associates all points with themselves. That's... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
It just points back at the point that you started off with, it associates all points with themselves. That's the identity function on x. If it, especially as it applies to the point a, if you apply it to some other point in x, it would just refer back to itself. Now that's the identity function on x. You could also hav... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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 drew b right there, then the y identity function, so this would be the identity function on y, applied to b, would just refer back to itself. And so that would be equal to b. This is the iden... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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 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 goi... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
They're actually 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, 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 he... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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. ... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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. So this implies that, and that implies this. So f is invertible. I'm kind of making a definition rig... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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 a function, we'll call it, I'll call it nothing just yet. I'll call it something in a second. And I'll write it as this f with this negative 1 superscript on it. So f is invertible if... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
If and only if there exists a function, we'll call it, I'll call it nothing just yet. I'll call it something in a second. And I'll write it as this f with this negative 1 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 tha... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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, let me do that in purple, such that if I apply f, remember f is just a mapping from x to y. So this function f inverse is going to be a mapping from y to x. So I'm saying that f is invert... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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, sorry, the composition of f with the inverse function has to be equal to the identity function over y... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
This is true, this has to be true, and f, the composition of f with the identity function, has to be, sorry, 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... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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. | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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 to x. So f inverse, if you give me some value in set y, I go to set x. So this guy's domain is this guy's c... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
So let's write it here. So f inverse is a mapping from y to x. So f inverse, if you give me some value in set y, I go to set x. So this guy's domain is this guy's codomain. And this guy's codomain is this guy's domain. Fair enough. But let's see what it's saying. | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
So this guy's domain is this guy's codomain. And this guy's codomain 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 to the identity matrix. So essentially it's saying, if I apply f to some value in x, if you think about what's this... | Introduction to the inverse of a function Matrix transformations Linear Algebra Khan Academy.mp3 |
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