Sentence stringlengths 126 2.7k | video_title stringlengths 60 104 |
|---|---|
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 careful, taking this composition, this transformation, and then taking that tra... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Let me be 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 we know that this is just the identity transformation on Rn. A... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
And I just rewrote this composition this way. I rewrote this composition this way. And the reason why I did this is 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 e... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
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 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 fo... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
C times T inverse times some vector A. And just like that, we've met our second condition for 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 used the fact that T was a linear transformation to get to the result for T inverse. So ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
The first condition was met up here. So now we know. And in both cases, we used the fact that T was a linear transformation to get to the result for T inverse. So now we know that if T is a linear transformation and T is invertible, then T inverse is also a linear transformation. Which might seem like a little nice thi... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So now we know that if T is a linear transformation and T is invertible, then T inverse is also a 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 inve... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
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 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 ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
And I haven't defined this well, how do you construct this A inverse matrix. But we know that it exists. We know this exists now because T is a linear transformation. And we could take it even a step further. We know by the definition of invertibility that the composition of T inverse with T is equal to the identity tr... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
And we could take it even a step further. We know by the definition of invertibility that the composition of T inverse with T is equal to the identity transformation on Rn. Well, what is the composition? 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 w... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
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, is 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 ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
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 the resulting transformation matrix of two composition transformations is equal to this matrix-matrix product. We got that a long time ago. In... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
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 going to be equal to the identity transformation on Rn applied to that vector x, which is equal to the identity matrix. The identity... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
The identity matrix applied to x, right? That is the n by n matrix. So 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 matr... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
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,... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
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, I Rn. So by the exact same argument, we know that when you go the other ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
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 an invertible matrix and its inverse, order doesn't matter. You ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
And this is, in general, terminology that you'll probably see in your mathematical careers. So let's say I have a function f. And it is a mapping from the set x to the set y. And we've drawn this diagram many times, but it never hurts to draw it again. So that is my set x or my domain. And then this is the set y over h... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
So that is my set x or my domain. And then this is the set y over here. Or the codomain. Remember, the codomain is the set that you're mapping to. You don't necessarily have to map to every element of the set or none of the elements of the set. This is just all of the elements, the set that you might map elements in yo... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Remember, the codomain is the set that you're mapping to. You don't necessarily have to map to every element of the set or none of the elements of the set. This is just all of the elements, the set that you might map elements in your codomain to. So let's see. If I have some element there, f will map it to some element... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
So let's see. If I have some element there, f will map it to some element in y, in my codomain. So the first idea or term I want to introduce you to is the idea of a function being surjective. And sometimes this is called onto. And a function is surjective or onto if for every element in your codomain. So let me write ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
And sometimes this is called onto. And a function is surjective or onto if for every element in your codomain. So let me write it this way. If for every, let's say, y that is a member of my codomain, there exists, that's the little shorthand notation for exists, there exists at least one x. That's a member of x such th... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
If for every, let's say, y that is a member of my codomain, there exists, that's the little shorthand notation for exists, there exists at least one x. That's a member of x such that, and I can write such that like that. Actually, let me just write the word out. Such that f of x is equal to y. So it's essentially sayin... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Such that f of x is equal to y. So it's essentially saying, look, you can pick any y here and every y here is being mapped to by at least one of the x's over here. So for example, actually let me draw a simpler example. Instead of drawing these blurbs, let's say that I have a set y that literally looks like this. Let's... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Instead of drawing these blurbs, let's say that I have a set y that literally looks like this. Let's say that a set y, I'll draw it very, and let's say it has four elements. It has the elements a, b, c, and d. This is my set y right there. And let's say my set x looks like that. And let's say it has the elements, I don... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
And let's say my set x looks like that. And let's say it has the elements, I don't know, 1, 2, 3, and 4. Now, in order for my function f to be surjective or onto it means that every one of these guys have to be able to be mapped to. So what does that mean? So if every one of these guys, let me just draw some examples. ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
So what does that mean? So if every one of these guys, let me just draw some examples. Let's say that this guy maps to that. Let's say that this guy maps to that. Let's say that this guy maps to that. And let's say, let me draw a fifth one right here. Let's say that both of these guys right here map to d. So f of 4 is ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Let's say that this guy maps to that. Let's say that this guy maps to that. And let's say, let me draw a fifth one right here. Let's say that both of these guys right here map to d. So f of 4 is d and f of 5 is d. This is an example of a surjective function. So these are the mappings of f right here. This function righ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Let's say that both of these guys right here map to d. So f of 4 is d and f of 5 is d. This is an example of a surjective function. So these are the mappings of f right here. This function right here is onto or surjective. Why is that? Because every element here is being mapped to. Now let me give you an example of a f... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Why is that? Because every element here is being mapped to. Now let me give you an example of a function that is not surjective. Let me add some more elements to y. Let's say y has another element here called e. Now all of a sudden, this is not surjective. And why is that? Because there's some element in y that is not ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Let me add some more elements to y. Let's say y has another element here called e. Now all of a sudden, this is not surjective. And why is that? Because there's some element in y that is not being mapped to. So if I tell you that f is a surjective function, it means that if you map all of these values, everything here ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Because there's some element in y that is not being mapped to. So if I tell you that f is a surjective function, it means that if you map all of these values, everything here is being mapped to by at least one element here. So everything could be kind of a one-to-one mapping, and I'll define that a little bit better in... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
So it could just be like that and like that. And you could even have at least one. So you could even have two things in here mapping to one thing in here. But the main requirement is that everything here does get mapped to. Another way to think about it is that if you take the image, so a surjective function, let me wr... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
But the main requirement is that everything here does get mapped to. Another way to think about it is that if you take the image, so a surjective function, let me write this here. So a surjective function, or let me write it this way. So if I say that f is surjective or onto, these are equivalent terms, that means that... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
So if I say that f is surjective or onto, these are equivalent terms, that means that the image of f, remember, the image was all of the values that f actually maps to. So that means that the image of f is equal to y. Now we learned before that your image doesn't have to equal your codomain, but if you have a surjectiv... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Everything in your codomain gets mapped to. Actually, another word for image is range. You could also say that your range of f is equal to y. Remember, the difference, and I drew this distinction when we first talked about functions, the distinction between a codomain and a range, a codomain is a set that you can map t... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Remember, the difference, and I drew this distinction when we first talked about functions, the distinction between a codomain and a range, a codomain is a set that you can map to. You don't have to map to everything. The range is a subset of your codomain that you actually do map to. If you were to evaluate the functi... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
If you were to evaluate the function at all of these points, the point that you actually map to is your range. And that's also called your image, and we use the word image is used more in a linear algebra context. But if your image or your range is equal to your codomain, if everything in your codomain does get mapped ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Now the next term I want to introduce you to is the idea of an injective function. Injective function. And this is sometimes called a one-to-one function. So let me draw my domain and codomain again. So let's say that that is my domain, and this is my codomain. So this is x and this is y. If I say that f is injective, ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
So let me draw my domain and codomain again. So let's say that that is my domain, and this is my codomain. So this is x and this is y. If I say that f is injective, or one-to-one, that implies that for every value that is mapped to, so let me write it this way, for every value that is mapped to, so let's say, I'll say ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
If I say that f is injective, or one-to-one, that implies that for every value that is mapped to, so let me write it this way, for every value that is mapped to, so let's say, I'll say it a couple of different ways, there is at most one x that maps to it. There is at most one x that maps to it. Or another way to say it... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
There might be no x's that map to it. So for example, you could have a little member of y right here that just never gets mapped to. Everyone else in y gets mapped to, but that guy never gets mapped to. So this would be a case where we don't have a surjective function. This is not on to, because this guy is a member of... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
So this would be a case where we don't have a surjective function. This is not on to, because this guy is a member of the codomain, but he's not a member of the image or the range, he doesn't get mapped to. But this would still be an injective function as long as every x gets mapped to a unique y. Now how can a functio... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
Now how can a function not be injective or one to one? And I think you get the idea when someone says one to one. Well if two x's here get mapped to the same y, or three get mapped to the same y, this would mean that we're not dealing with an injective or a one to one function. So that's all it means. Let me draw anoth... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
So that's all it means. Let me draw another example here. So if I take, let's actually go back to this example right here. When I added this e here, we said this is not surjective anymore, because every one of these guys is not being mapped to. Is this an injective function? Well no, because I have f of 5 and f of 4 bo... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
When I added this e here, we said this is not surjective anymore, because every one of these guys is not being mapped to. Is this an injective function? Well no, because I have f of 5 and f of 4 both map to d. So this is what breaks its one to oneness, or its injectiveness. This is what breaks its surjectiveness. If I ... | Surjective (onto) and injective (one-to-one) functions Linear Algebra Khan Academy.mp3 |
In this video, I want to kind of go back to basics and just give you a lot of examples and give you a more tangible sense for what vectors are and how we operate with them. So let me define a couple of vectors here. And most of my vectors I'm going to do in this video are going to be an r2. And that's because they're e... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And that's because they're easy to draw. r2, remember, r2 was a set of all two tuples, ordered two tuples, where each of the numbers, so you could have x1, and let me see, my 1 looks like a comma, x1 and x2, where each of these are real numbers. So each of them, x1 is a member of the reals, x1, and x2 is a member of th... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And just to give you a sense of what that means, if this right here is my coordinate axes and I wanted to plot all my x1's, x2's, you could view this as the first coordinate. We always imagine that as our x-axis. And then our second coordinate, we plot it on the vertical axis that traditionally is our y-axis, but we'll... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
You could visually represent all of r2 by literally every single point on this plane if we were to continue off in infinity in every direction. That's what r2 is. r1 would just be points just along one of these number lines, that would be r1. So you could immediately see that r2 is kind of a bigger space. But anyway, I... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So you could immediately see that r2 is kind of a bigger space. But anyway, I said that I wouldn't be too abstract, I would show you examples. So let's get some vectors going in r2. So let me define my vector a. I'll make it nice and bold. My vector a is equal to, I'll make some numbers up, negative 1, 2, and my vector... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let me define my vector a. I'll make it nice and bold. My vector a is equal to, I'll make some numbers up, negative 1, 2, and my vector b, make it nice and bold, let me make that, I don't know, 3, 3, 1. Those are my two vectors. Now let's just add them up and see what we get. Just based on my definition of vector ad... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now let's just add them up and see what we get. Just based on my definition of vector addition, and I'll just stay in one color for now, just so I don't have to keep switching back and forth. So a, nice deep a, plus bolded b is equal to, I just add up each of those terms, negative 1 plus 3, and then 2 plus 1. That was ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
That was my definition of vector addition. So that is going to be equal to 2 and 3. Fair enough. This just came out of my definition of vector addition. But how can we represent this vector? So we already know that if we have coordinates, if I have the coordinate, and this is just a convention, it's just the way that w... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
This just came out of my definition of vector addition. But how can we represent this vector? So we already know that if we have coordinates, if I have the coordinate, and this is just a convention, it's just the way that we do it, the way we visualize things. If I wanted to plot the point 1, 1, I go to my coordinate a... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
If I wanted to plot the point 1, 1, I go to my coordinate axes, the first point I go along the horizontal, what we traditionally call our x-axis, and I go 1 in that direction, and then the convention is the second point, I go 1 in the vertical direction. So the point 1, 1, I would, oh sorry, let me be very clear, this ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
That's just the standard convention. Now, our convention for representing vectors are, you might be tempted to say, oh, maybe I just represent this vector at the point minus 1, 2. And on some level you can do that, and I'll show you in a second. But the convention for vectors is that you can start at any point. Let's s... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
But the convention for vectors is that you can start at any point. Let's say we're dealing with two-dimensional vectors. You can start at any point in R2. So let's say that you're starting at the point x1 and x2. This could be any point in R2. To represent this vector, what we do is we draw a line from that point to th... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let's say that you're starting at the point x1 and x2. This could be any point in R2. To represent this vector, what we do is we draw a line from that point to the point x1, and let me call this, let's say that we wanted to draw a. So x1 minus 1, so I'm representing a. So I want to represent the vector a. x1 minus 1... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So x1 minus 1, so I'm representing a. So I want to represent the vector a. x1 minus 1, and then x1 plus 2. Now, if that seems confusing to you, when I draw it, it'll be very obvious. So let's say I just want to start at the point, let's just say for quirky reasons, I just pick a random point here. I just pick a point, ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let's say I just want to start at the point, let's just say for quirky reasons, I just pick a random point here. I just pick a point, that one right there. That's my starting point. So minus 4, 4. That's minus 4, 4. Now, if I want to represent my vector a, what I just said is that I add the first term in vector a to... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So minus 4, 4. That's minus 4, 4. Now, if I want to represent my vector a, what I just said is that I add the first term in vector a to my first coordinate, so x1 plus minus 1, or x1 minus 1. So my new one is going to be, so this is my x1 minus 4. So now it's going to be, so let's see, I'm starting at the point minus 4... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So my new one is going to be, so this is my x1 minus 4. So now it's going to be, so let's see, I'm starting at the point minus 4, 4. If I want to represent a, what I do is I draw an arrow to minus 4 plus this first term, minus 1, and then 4 plus the second term, 4 plus 2. And so this is what? This is minus 5, 6. So I g... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And so this is what? This is minus 5, 6. So I go to minus 5, 6. So I go to that point right there, and I just draw a line. So my vector will look like this. I draw a line from there to there, and I draw an arrow at the end point. So that's one representation of the vector minus 1, 2. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So I go to that point right there, and I just draw a line. So my vector will look like this. I draw a line from there to there, and I draw an arrow at the end point. So that's one representation of the vector minus 1, 2. Actually, let me do it a little bit better. Because minus 5 is actually more, it's a little closer ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So that's one representation of the vector minus 1, 2. Actually, let me do it a little bit better. Because minus 5 is actually more, it's a little closer to right here. Minus 5, 6 is right there. So I draw my vector like that. But remember, this point minus 4, 4 was an arbitrary place to draw my vector. I could have st... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Minus 5, 6 is right there. So I draw my vector like that. But remember, this point minus 4, 4 was an arbitrary place to draw my vector. I could have started at this point here. I could have started at the point 4, 6 and done the same thing. I could have gone minus 1 in the horizontal direction. That's my movement in th... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
I could have started at this point here. I could have started at the point 4, 6 and done the same thing. I could have gone minus 1 in the horizontal direction. That's my movement in the horizontal direction. And then plus 2 in the vertical direction. So I could have drawn, so minus 1 in the horizontal, and then plus 2 ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
That's my movement in the horizontal direction. And then plus 2 in the vertical direction. So I could have drawn, so minus 1 in the horizontal, and then plus 2 in the vertical, gets me right there. So I could have just as easily drawn my vector like that. These are both interpretations of the same vector a. And I shoul... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So I could have just as easily drawn my vector like that. These are both interpretations of the same vector a. And I should draw them in the color of vector a. So vector a was this light blue color right there. So this is vector a. Sometimes there'll be a little arrow notation over the vector, but either of those vecto... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So vector a was this light blue color right there. So this is vector a. Sometimes there'll be a little arrow notation over the vector, but either of those vectors. I could draw an infinite number of vector a's. I could draw vector a here. I could draw it like that. And vector a, it goes back 1 and up 2. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
I could draw an infinite number of vector a's. I could draw vector a here. I could draw it like that. And vector a, it goes back 1 and up 2. So vector a could be right there. Similarly, vector b. What does vector b do? | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And vector a, it goes back 1 and up 2. So vector a could be right there. Similarly, vector b. What does vector b do? I could pick some arbitrary point for vector b. It goes to the right 3. So it goes to the right 1, 2, 3. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
What does vector b do? I could pick some arbitrary point for vector b. It goes to the right 3. So it goes to the right 1, 2, 3. And then it goes up 1. So vector b, one representation of vector b looks like this. Another representation, I could start. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So it goes to the right 1, 2, 3. And then it goes up 1. So vector b, one representation of vector b looks like this. Another representation, I could start. I could do it right here. I could start it right here. I could go to the right 3, 1, 2, 3, and then up 1. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Another representation, I could start. I could do it right here. I could start it right here. I could go to the right 3, 1, 2, 3, and then up 1. This would be another representation of my vector b. There's an infinite number of representations of them, but the convention is to often put them in what's called the standa... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
I could go to the right 3, 1, 2, 3, and then up 1. This would be another representation of my vector b. There's an infinite number of representations of them, but the convention is to often put them in what's called the standard position. And that's to start them off at 0, 0. So your initial point, so let me write this... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And that's to start them off at 0, 0. So your initial point, so let me write this down. Standard position is just to start the vectors at 0, 0 and then draw them. So vector a in standard position, I would start at 0, 0 like that. And I would go back 1 and then up 2. So this is vector a in standard position right there.... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So vector a in standard position, I would start at 0, 0 like that. And I would go back 1 and then up 2. So this is vector a in standard position right there. And then vector b in standard position. Let me write that. That's a. And then vector b in standard position is 3. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then vector b in standard position. Let me write that. That's a. And then vector b in standard position is 3. Go to 3 right and then up 1. But these are the vectors in standard position, but any of these other things we drew are just as valid. Now let's see if we can get an interpretation of what happened when we a... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then vector b in standard position is 3. Go to 3 right and then up 1. But these are the vectors in standard position, but any of these other things we drew are just as valid. Now let's see if we can get an interpretation of what happened when we added a plus b. Well, if I draw that vector in standard position, I ju... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now let's see if we can get an interpretation of what happened when we added a plus b. Well, if I draw that vector in standard position, I just calculated, and I go 2, 3. So I go to the right 2 and I go up 3. So if I just draw it in standard position, it looks like this. That vector right there. And at first when you l... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So if I just draw it in standard position, it looks like this. That vector right there. And at first when you look at it, so this vector right here is the vector a plus b in standard position. When you draw it like that, it's not clear what the relationship is when we added a and b. But to see the relationship, what yo... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
When you draw it like that, it's not clear what the relationship is when we added a and b. But to see the relationship, what you do is you put a and b head to tail. So what does that mean? You put the tail end of b to the front end of a. So you can remember, all of these are valid representations of b. All of the repre... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
You put the tail end of b to the front end of a. So you can remember, all of these are valid representations of b. All of the representations of the vector b, they're all parallel to each other, but they can start from anywhere. So another equally valid representation of vector b is to start at this point right here, k... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So another equally valid representation of vector b is to start at this point right here, kind of the end point of vector a in standard position, and then draw vector b starting from there. So you go 3 to the right. So you go 1, 2, 3, and then you go up 1. So vector b could also be drawn just like that. And then you sh... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So vector b could also be drawn just like that. And then you should see something interesting had happened. And remember, this vector b representation is not in standard position, but it's just an equally valid way to represent my vector. Now what do you see? When I add a, which is right here, to b, what do I get if I ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now what do you see? When I add a, which is right here, to b, what do I get if I connect the starting point of a with the end point of b? I get the addition. I have added the two vectors. And I could have done that anywhere. I could have started with a here, and then I could have done the end point of, I could have sta... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
I have added the two vectors. And I could have done that anywhere. I could have started with a here, and then I could have done the end point of, I could have started b here and gone 3 to the right, 1, 2, 3, and then up 1. And I could have drawn b right there like that. And then if I were to add a plus b, I go to the s... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And I could have drawn b right there like that. And then if I were to add a plus b, I go to the starting point of a and then the end point of b. And that should also be the visual representation of a plus b. And just to make sure it confirms with this number, what I did here is I went 2 to the right, 1, 2, and then I w... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And just to make sure it confirms with this number, what I did here is I went 2 to the right, 1, 2, and then I went 3 up, 1, 2, 3. And I got a plus b. Now let's think about what happens when we scale our vectors, when we multiply it by times some scalar factor. So let me pick new vectors. Those have gotten monotonous. ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let me pick new vectors. Those have gotten monotonous. Let me define vector v. v for vector. Let's say that it is equal to 1, 2. So if I just wanted to draw vector v in standard position, I would just go 1 to the horizontal and then 2 to the vertical. That's the vector in standard position. If I wanted to do it in a... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's say that it is equal to 1, 2. So if I just wanted to draw vector v in standard position, I would just go 1 to the horizontal and then 2 to the vertical. That's the vector in standard position. If I wanted to do it in a non-standard position, I could do it right here, 1 to the right, up 2, just like that. Equally ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
If I wanted to do it in a non-standard position, I could do it right here, 1 to the right, up 2, just like that. Equally valid way of drawing vector v. Equally valid way of doing it. Now what happens if I multiply vector v? What if I have 2 times v? 2 times my vector v is now going to be equal to 2 times each of these ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
What if I have 2 times v? 2 times my vector v is now going to be equal to 2 times each of these terms. So it's going to be 2 times 1, which is 2, and then 2 times 2, which is 4. Now what does 2 times vector v look like? Well, let me just start from an arbitrary position. Let me just start right over here. So I'm going ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now what does 2 times vector v look like? Well, let me just start from an arbitrary position. Let me just start right over here. So I'm going to go 2 to the right, 1, 2, and I go up 4. 1, 2, 3, 4. So this is what 2 times vector v looks like. This is 2 times my vector v. And if you look at it, it's pointing in the exact... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So I'm going to go 2 to the right, 1, 2, and I go up 4. 1, 2, 3, 4. So this is what 2 times vector v looks like. This is 2 times my vector v. And if you look at it, it's pointing in the exact same direction, but now it's twice as long. And that makes sense, because we scaled it by a factor of 2. When you multiply it by... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
This is 2 times my vector v. And if you look at it, it's pointing in the exact same direction, but now it's twice as long. And that makes sense, because we scaled it by a factor of 2. When you multiply it by a scalar, you're not changing its direction. Its direction is the exact same thing as it was before. You're just... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Its direction is the exact same thing as it was before. You're just scaling it by that amount. And I could draw this anywhere. I could have drawn it right here. I could have drawn 2v right on top of v, and then you would have seen it, and I don't want to cover it, you would have seen that it's exactly, in this case, wh... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
I could have drawn it right here. I could have drawn 2v right on top of v, and then you would have seen it, and I don't want to cover it, you would have seen that it's exactly, in this case, when I drew it in standard position, it's collinear. It's along the same line. It's just twice as far. It's just twice as long, b... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.