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It's just twice as far. It's just twice as long, but they have the exact same direction. Now what happens if I were to multiply minus 4 times our vector v? Well, then that will be equal to minus 4 times 1, which is minus 4, and then minus 4 times 2, which is minus 8. So this is my new vector. Minus 4 minus 8. This is m... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, then that will be equal to minus 4 times 1, which is minus 4, and then minus 4 times 2, which is minus 8. So this is my new vector. Minus 4 minus 8. This is minus 4 times our vector v. So let's just start at some arbitrary point. Let's just do it in standard position. So you go to the left 4, and then down 8. It'... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
This is minus 4 times our vector v. So let's just start at some arbitrary point. Let's just do it in standard position. So you go to the left 4, and then down 8. It'll look like that. So this new vector is going to go, and this is going to look like this. I want to make sure I can draw a relatively straight line. There... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
It'll look like that. So this new vector is going to go, and this is going to look like this. I want to make sure I can draw a relatively straight line. There you go. So this is minus 4 times our vector v. I'll do an arrow on it to make sure you know it's a vector. Now what happened? Well, we're kind of in the same dir... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
There you go. So this is minus 4 times our vector v. I'll do an arrow on it to make sure you know it's a vector. Now what happened? Well, we're kind of in the same direction. Actually, we're in the exact opposite direction, but we're still along the same line. We're just in the exact opposite direction. And it's this n... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, we're kind of in the same direction. Actually, we're in the exact opposite direction, but we're still along the same line. We're just in the exact opposite direction. And it's this negative right there that flipped us around. If we just multiplied negative 1 times this, we would have just flipped around to right ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And it's this negative right there that flipped us around. If we just multiplied negative 1 times this, we would have just flipped around to right there. But we multiplied it by negative 4. So we scale it by 4, so you make it 4 times as long, and then it's negative. So then it flips around. It flips backwards. So now t... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So we scale it by 4, so you make it 4 times as long, and then it's negative. So then it flips around. It flips backwards. So now that we have that notion, we can start understanding the idea of subtracting vectors. Let me make up two new vectors right now. Let's say my vector x is equal to 2, 4. And let's say I have a ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So now that we have that notion, we can start understanding the idea of subtracting vectors. Let me make up two new vectors right now. Let's say my vector x is equal to 2, 4. And let's say I have a vector y. Make it nice and bold. And then that is equal to negative 1 minus 2. And I want to think about the notion of wha... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And let's say I have a vector y. Make it nice and bold. And then that is equal to negative 1 minus 2. And I want to think about the notion of what x minus y is equal to. Well, we can say that this is the same thing as x plus minus 1 times our vector y. So x plus minus 1 times our vector y. Now we can use our definition... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And I want to think about the notion of what x minus y is equal to. Well, we can say that this is the same thing as x plus minus 1 times our vector y. So x plus minus 1 times our vector y. Now we can use our definitions. We know how to multiply by a scalar. So we'll say that this is equal to our x vector is 2, 4. And t... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now we can use our definitions. We know how to multiply by a scalar. So we'll say that this is equal to our x vector is 2, 4. And then what's minus 1 times y? So minus 1 times y is 1. And then minus 1 times minus 2 is 2. So x minus y is going to be these two vectors added to each other. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then what's minus 1 times y? So minus 1 times y is 1. And then minus 1 times minus 2 is 2. So x minus y is going to be these two vectors added to each other. I'm just adding the minus of y. This right here is minus vector y. So this x minus y is going to be equal to 3 and 6. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So x minus y is going to be these two vectors added to each other. I'm just adding the minus of y. This right here is minus vector y. So this x minus y is going to be equal to 3 and 6. So let's see what that looks like when we visually represent them. Our vector x was 2, 4. So 2, 4. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So this x minus y is going to be equal to 3 and 6. So let's see what that looks like when we visually represent them. Our vector x was 2, 4. So 2, 4. In standard position, it looks like this. That's my vector x. And then vector y in standard position. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So 2, 4. In standard position, it looks like this. That's my vector x. And then vector y in standard position. Let me do it in a different color. I'll do y in green. Vector y is minus 1 minus 2. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then vector y in standard position. Let me do it in a different color. I'll do y in green. Vector y is minus 1 minus 2. So minus 1 minus 2. So minus 1 minus 2. It looks just like this. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Vector y is minus 1 minus 2. So minus 1 minus 2. So minus 1 minus 2. It looks just like this. And actually I ended up inadvertently doing collinear vectors. But hey, this is interesting too. So this is vector y. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
It looks just like this. And actually I ended up inadvertently doing collinear vectors. But hey, this is interesting too. So this is vector y. So then what's their difference? It says 3, 6. So it's the vector 3, 6. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So this is vector y. So then what's their difference? It says 3, 6. So it's the vector 3, 6. So it's this vector. Let me draw it someplace else. If I start here, I go 1, 2, 3. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So it's the vector 3, 6. So it's this vector. Let me draw it someplace else. If I start here, I go 1, 2, 3. And then I go up 6. So then up 6. It's a vector that looks like this. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
If I start here, I go 1, 2, 3. And then I go up 6. So then up 6. It's a vector that looks like this. That's the difference between the two vectors. So at first you say, well, this is x minus y. You're like, hey, how is this the difference of these two? | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
It's a vector that looks like this. That's the difference between the two vectors. So at first you say, well, this is x minus y. You're like, hey, how is this the difference of these two? Well, if you overlay this, if you just shift this and over this, you could actually just start here and go straight up. And you'll s... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
You're like, hey, how is this the difference of these two? Well, if you overlay this, if you just shift this and over this, you could actually just start here and go straight up. And you'll see that it's really the difference between the endpoints. You're kind of connecting the endpoints. But I actually didn't want to ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
You're kind of connecting the endpoints. But I actually didn't want to draw collinear vectors. So let me do another example. Although that one's kind of interesting. You often don't see that one in a book. Let me define vector x, in this case, to be 2, 3. And let me define vector y to be minus 4, minus 2. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Although that one's kind of interesting. You often don't see that one in a book. Let me define vector x, in this case, to be 2, 3. And let me define vector y to be minus 4, minus 2. So what would be x in standard position? It would be 2, 3. It'd look like that. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And let me define vector y to be minus 4, minus 2. So what would be x in standard position? It would be 2, 3. It'd look like that. That is our vector x if we start at the origin. So this is x. And then what does vector y look like? | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
It'd look like that. That is our vector x if we start at the origin. So this is x. And then what does vector y look like? I'll do y in orange. Minus 4, minus 2. So vector y looks like this. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then what does vector y look like? I'll do y in orange. Minus 4, minus 2. So vector y looks like this. Now, what is x minus y? Well, we could view this 2 plus minus 1 times this. Or we could just say 2 minus minus 4. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So vector y looks like this. Now, what is x minus y? Well, we could view this 2 plus minus 1 times this. Or we could just say 2 minus minus 4. I think you get the idea now. But we just did it the first way the last time. Because I wanted to go for my basic definitions of scalar multiplication. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Or we could just say 2 minus minus 4. I think you get the idea now. But we just did it the first way the last time. Because I wanted to go for my basic definitions of scalar multiplication. So x minus y is just going to be equal to 2 plus minus 1 times minus 4, or 2 minus minus 4. That's the same thing as 2 plus 4. So ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Because I wanted to go for my basic definitions of scalar multiplication. So x minus y is just going to be equal to 2 plus minus 1 times minus 4, or 2 minus minus 4. That's the same thing as 2 plus 4. So it's 6. And then it's 3 minus minus 2. So it's 5. So the difference between the two is the vector 6, 5. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So it's 6. And then it's 3 minus minus 2. So it's 5. So the difference between the two is the vector 6, 5. So you could draw it out here again. So you could go, to add 6 to 4, you go up there. And then to 5, you'd go like that. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So the difference between the two is the vector 6, 5. So you could draw it out here again. So you could go, to add 6 to 4, you go up there. And then to 5, you'd go like that. So the vector would look something like this. It shouldn't curve like that. So that's x minus y. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then to 5, you'd go like that. So the vector would look something like this. It shouldn't curve like that. So that's x minus y. But if we drew them between, like in the last example, I showed that you could draw it between their two heads. So if you do it here, what does it look like? Well, if you start at this poi... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So that's x minus y. But if we drew them between, like in the last example, I showed that you could draw it between their two heads. So if you do it here, what does it look like? Well, if you start at this point right there, and you go 6 to the right, and then up 5, you end up right there. So the difference between the... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, if you start at this point right there, and you go 6 to the right, and then up 5, you end up right there. So the difference between the two vectors looks like that. Which kind of should make sense intuitively. x minus y. That's the difference between the two vectors. You can view the difference as how do you get ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
x minus y. That's the difference between the two vectors. You can view the difference as how do you get from one vector to another vector, right? Like, let's go back to our second grade world of just scalars. If I say what 7 minus 5 is, and you say it's equal to 2, well, that just tells you that 5 plus 2 is equal to 7.... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Like, let's go back to our second grade world of just scalars. If I say what 7 minus 5 is, and you say it's equal to 2, well, that just tells you that 5 plus 2 is equal to 7. Or the difference between 5 and 7 is 2. And here you're saying, look, the difference between x and y is this vector right there. It's equal to th... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And here you're saying, look, the difference between x and y is this vector right there. It's equal to that vector right there. Or you could say, look, if I take 5 and add 2, I get 7. Or you could say, look, if I take vector y and I add vector x minus y, then I get vector x. Now, let's do something else that's interest... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Or you could say, look, if I take vector y and I add vector x minus y, then I get vector x. Now, let's do something else that's interesting. Let's do what y minus x is equal to. y minus x. What is that equal to? Do another color right here. Well, we'll take minus 4 minus 2, which is minus 6. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
y minus x. What is that equal to? Do another color right here. Well, we'll take minus 4 minus 2, which is minus 6. And then you have minus 2 minus 3. It's minus 5. So y minus x is going to be, let's see, if we start here, we're going to go down 6, 1, 2, 3, 4, 5, 6, and then back 5. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, we'll take minus 4 minus 2, which is minus 6. And then you have minus 2 minus 3. It's minus 5. So y minus x is going to be, let's see, if we start here, we're going to go down 6, 1, 2, 3, 4, 5, 6, and then back 5. So back 2, 4, 5. So y minus x looks like this. It's really the exact same vector. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So y minus x is going to be, let's see, if we start here, we're going to go down 6, 1, 2, 3, 4, 5, 6, and then back 5. So back 2, 4, 5. So y minus x looks like this. It's really the exact same vector. Remember, it doesn't matter where we start. It's just pointing in the opposite direction. So if we shifted it here, I c... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
It's really the exact same vector. Remember, it doesn't matter where we start. It's just pointing in the opposite direction. So if we shifted it here, I could draw it right on top of this. It would be the exact as x minus y, but just in the opposite direction. Which is just a general good thing to know. So you can kind... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So if we shifted it here, I could draw it right on top of this. It would be the exact as x minus y, but just in the opposite direction. Which is just a general good thing to know. So you can kind of view them as the negatives of each other. And actually, let me make that point very clear. We drew y. y I drew before. Ac... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
So you can kind of view them as the negatives of each other. And actually, let me make that point very clear. We drew y. y I drew before. Actually, let me draw x. x we could draw as 2, 3. So you go to the right 2 and then up 3. I've done this before. This is x in non-standard position. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Actually, let me draw x. x we could draw as 2, 3. So you go to the right 2 and then up 3. I've done this before. This is x in non-standard position. That's x as well. What is negative x? Negative x is minus 2 minus 3. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
This is x in non-standard position. That's x as well. What is negative x? Negative x is minus 2 minus 3. So if I were to start here, I'd go minus 2 and then I'd go minus 3. So minus x would look just like this. It looks just like x. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Negative x is minus 2 minus 3. So if I were to start here, I'd go minus 2 and then I'd go minus 3. So minus x would look just like this. It looks just like x. It's parallel. It has the same magnitude. It's just pointing in the exact opposite direction. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
It looks just like x. It's parallel. It has the same magnitude. It's just pointing in the exact opposite direction. This is just a good thing to kind of really get seared into your brain is to have an intuition for these things. Now, just to kind of finish up this kind of idea of adding and subtracting vectors, let me ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
It's just pointing in the exact opposite direction. This is just a good thing to kind of really get seared into your brain is to have an intuition for these things. Now, just to kind of finish up this kind of idea of adding and subtracting vectors, let me just do everything I did so far was in an R2. But I want to show... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
But I want to show you that we can generalize them. And we can even generalize them to vector spaces that aren't normally intuitive for us to actually visualize. So let me define a couple of vectors. Let me define vector a to be equal to 0, minus 1, 2, and 3. Let me define vector b to be equal to 4, minus 2, 0, 5. We c... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let me define vector a to be equal to 0, minus 1, 2, and 3. Let me define vector b to be equal to 4, minus 2, 0, 5. We can do the same addition and subtraction operations with them. It's just it'll be hard to visualize. But we can keep them in just vector form so that it's still useful to think in four dimensions. So i... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
It's just it'll be hard to visualize. But we can keep them in just vector form so that it's still useful to think in four dimensions. So if I were to say 4 times a, this is the vector a, minus 2 times b, what is this going to be equal to? And this is a vector. What is this going to be equal to? Well, we could rewrite t... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And this is a vector. What is this going to be equal to? Well, we could rewrite this as 4 times this whole column vector, 0, minus 1, 2, and 3, minus 2 times b, minus 2 times 4, minus 2, 0, 5. And what is this going to be equal to? This term right here, 4 times this, you're going to get 4 times 0 is 0, minus 4, 8. 4 ti... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And what is this going to be equal to? This term right here, 4 times this, you're going to get 4 times 0 is 0, minus 4, 8. 4 times 2 is 8, 4 times 3 is 12. And then minus, I'll do it in yellow, minus 2 times 4 is 8, 2 times minus 2 is minus 4, 2 times 0 is 0, 2 times 5 is 10. Now this isn't a good part of my board, so ... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then minus, I'll do it in yellow, minus 2 times 4 is 8, 2 times minus 2 is minus 4, 2 times 0 is 0, 2 times 5 is 10. Now this isn't a good part of my board, so let me just, it doesn't write well right over there. I haven't figured out the problem, but if I were to just write it over here, what do we get? With 0 min... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
With 0 minus 8, minus 4, minus 4, minus negative 4. So that's minus 4 plus 4, so that's 0. 8 minus 0 is 8, minus 4, minus 4. 8, 12 minus, what was this? I can't even read it, what it says. This was a 10. Oh, now you can see it again. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
8, 12 minus, what was this? I can't even read it, what it says. This was a 10. Oh, now you can see it again. Something, it's very bizarre. 2 times 5 is 10. So it's 12 minus 10, so it's 2. | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
Oh, now you can see it again. Something, it's very bizarre. 2 times 5 is 10. So it's 12 minus 10, so it's 2. So when we take this vector and multiply it by 4 and subtract 2 times this vector, we just get this vector. And even though you can't represent this in kind of an easy, kind of graphable format, this is a useful... | Vector examples Vectors and spaces Linear Algebra Khan Academy.mp3 |
It's called a column space. And you could probably guess what it means just based on what it's called. But let's say I have some matrix A. Let's say it's an m by n matrix. So I can write my matrix A, and we've seen this multiple times. I can write it as a collection of column vectors. So this first one, second one, and... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's say it's an m by n matrix. So I can write my matrix A, and we've seen this multiple times. I can write it as a collection of column vectors. So this first one, second one, and I'll have n of them. How do I know that I have n of them? Because I have n columns. And each of these column vectors are going to have how... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So this first one, second one, and I'll have n of them. How do I know that I have n of them? Because I have n columns. And each of these column vectors are going to have how many components? So v1, v2, all the way to vn. This matrix has m rows, right? So each of these guys are going to have m components. | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
And each of these column vectors are going to have how many components? So v1, v2, all the way to vn. This matrix has m rows, right? So each of these guys are going to have m components. So they're all members of Rm. So the column space is defined as all of the possible linear combinations of these column vectors. So t... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So each of these guys are going to have m components. So they're all members of Rm. So the column space is defined as all of the possible linear combinations of these column vectors. So the column space of A, this is my matrix A, the column space of that is all the linear combinations of these column vectors. Well, wha... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So the column space of A, this is my matrix A, the column space of that is all the linear combinations of these column vectors. Well, what's all of the linear combinations of a set of vectors? It's the span of those vectors. So it's the span of vector 1, vector 2, all the way to vector n. And we've done it before when ... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So it's the span of vector 1, vector 2, all the way to vector n. And we've done it before when we first talked about span and subspaces, but it's pretty easy to show that the span of any set of vectors is a legitimate subspace. It definitely contains the 0 vector. If you multiply all of these guys by 0, which is a vali... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's say I have some vector A that is a member of the column space of A. That means it can be represented as some linear combination. So A is equal to c1 times vector 1 plus c2 times vector 2, all the way to cn times vector n. Now the question is, is this closed under multiplication? Is if I multiply A times some new,... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Is if I multiply A times some new, let me call it, let me say I multiply it times some scalar s. I'm just picking a random letter. So s times A. Is this in my span? Well, s times A would be equal to sc1v1 plus sc2v2, all the way to scnvn, which is once again just a linear combination of these column vectors. So this sa... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, s times A would be equal to sc1v1 plus sc2v2, all the way to scnvn, which is once again just a linear combination of these column vectors. So this sa would clearly be a member of the column space of A. And then finally, to make sure it's a valid subspace, and this actually doesn't apply just to column spaces. Thi... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
This applies to any span. This is actually a review of what we've done in the past. We just have to make sure it's closed under addition. So let's say A is a member of our column space. Let's say B is also a member of our column space, or our span of all of these column vectors. Then B could be rewritten as, I don't kn... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let's say A is a member of our column space. Let's say B is also a member of our column space, or our span of all of these column vectors. Then B could be rewritten as, I don't know, let me say b1 times v1 plus b2 times v2, all the way to bn times vn. And my question is, is A plus B a member of our span, of our colu... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
And my question is, is A plus B a member of our span, of our column space, the span of these vectors? Well, sure. What's A plus B? A plus B is equal to c1 plus b1 times v1 plus c2 plus b2 times v2. I'm just literally adding this term to that term to get that term, this term to this term to get this term, and then it go... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
A plus B is equal to c1 plus b1 times v1 plus c2 plus b2 times v2. I'm just literally adding this term to that term to get that term, this term to this term to get this term, and then it goes all the way to bn plus cn times vn, which is clearly just another linear combination of these guys. So this guy is definitely wi... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So the span of, and this, what I just did, it doesn't have to be unique to a matrix. I mean, a matrix is just really just a way of writing a set of column vectors. So this applies to any span. So this is clearly a valid subspace. So the column space of A is clearly a valid subspace. Now let's think about other ways we ... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So this is clearly a valid subspace. So the column space of A is clearly a valid subspace. Now let's think about other ways we can interpret this notion of a column space. Let's think about it in terms of the expression, let me get a good color. If I were to multiply my, let's think about this. Let's think about the se... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's think about it in terms of the expression, let me get a good color. If I were to multiply my, let's think about this. Let's think about the set of all the values of, if I take my m by n matrix A and I multiply it by any vector x where x is a member of, remember, x has to be a member of Rn, it has to have n compon... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So x has to be a member of Rn. Let's think about what this means. This is the set, if this says, look, I can take any member, any n component vector and multiply it by a, and I care about all of the possible products that this could equal, all the possible values of Ax when I can pick and choose any possible x from Rn.... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's think about what that means. Ax, if I write a like that and if I write x like this, let me write it a little bit better, let me write x like this, x1, x2, all the way to xn. What is Ax? Well, Ax could be rewritten as x1, and we've seen this before, Ax is equal to x1 times v1 plus x2 times v2 all the way to plus x... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, Ax could be rewritten as x1, and we've seen this before, Ax is equal to x1 times v1 plus x2 times v2 all the way to plus xn times vn. We've seen this multiple times. This comes out of our definition of matrix vector products. Now, if Ax is equal to this, and I'm essentially saying that I can pick any vector x in ... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now, if Ax is equal to this, and I'm essentially saying that I can pick any vector x in Rn, I'm saying that I can pick all possible values of the entries here, all possible real values and all possible combinations of them. So what is this equal to? What is the set of all possible? So I could rewrite this statement her... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So I could rewrite this statement here as the set of all possible x1, v1, plus x2, v2, all the way to xn, vn, where x1, x2, all the way to xn are a member of the real numbers. That's all I'm saying here. This statement is the equivalent of this. When I say that the vector x can be any member of Rn, I'm saying that its ... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
When I say that the vector x can be any member of Rn, I'm saying that its components can be any members of the real numbers. So if I just take the set of all of the, essentially, the combinations of these column vectors where their real numbers, where their coefficients are members of the real numbers, what am I doing?... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So this is equal to the span of the v1, v2, all the way to vn, which is the same exact same thing as the column space of A. So the column space of A, you could say, hey, what are all of the possible vectors, or the set of all vectors I can create by taking linear combinations of these guys, or the span of these guys? O... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let's think about it this way. Let's say that I were to tell you that I need to solve the equation Ax is equal to, let me say, well, the convention is to write a b there, but let me put a special b there. Let me put b1. Let's say that I need to solve this equation Ax is equal to b1. And then I were to tell you, let'... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's say that I need to solve this equation Ax is equal to b1. And then I were to tell you, let's say that I were to figure out the column space of A, and I say b1 is not a member of the column space of A. Now what does that tell me? That tells me that this right here can never take on the value b1, right? Because all... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
That tells me that this right here can never take on the value b1, right? Because all of the values that this can take on is the column space of A. So if b1 is not in this, it means that this cannot take on the value of b1. So this would imply that this equation we're trying to set up, Ax is equal to b1, has no solutio... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So this would imply that this equation we're trying to set up, Ax is equal to b1, has no solution. If it had a solution, if it had some solution, so let's say that Ax equals b2. So let's say Ax equals b2 has at least one solution. Has at least one solution. What does this mean? Well, that means that this, for a particu... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
Has at least one solution. What does this mean? Well, that means that this, for a particular x, or maybe for many different x's, you can definitely achieve this value. For there are some x's that when you multiply it by a, you definitely are able to get this value. So this implies that b2 is definitely a member of the ... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
For there are some x's that when you multiply it by a, you definitely are able to get this value. So this implies that b2 is definitely a member of the column space of A. Some of this stuff, on some level, it's almost obvious. This comes out of the definition of the column space, the column space is all of the linear c... | Column space of a matrix Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let's actually use that method in this video right here. So I'm going to use the same matrix that we started off with in the last video. And it seems like a fairly good matrix. We know that its reduced row echelon form is the identity matrix, so we know it's invertible. So let's find its inverse. So the technique is... | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
We know that its reduced row echelon form is the identity matrix, so we know it's invertible. So let's find its inverse. So the technique is pretty straightforward. You literally just apply the same transformations you would apply to this guy to get you to the identity matrix. You would apply those same transformations... | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
You literally just apply the same transformations you would apply to this guy to get you to the identity matrix. You would apply those same transformations to the identity matrix. And that's because the collection of those transformations, if you represent them as matrices, are really just the inverse of this guy. So l... | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
So let's just do it. So I'll create an augmented matrix here. Maybe I'll do it right here. Let me make it a little bit neater. So first I'll write a. So it's 1, minus 1, 1. And then minus 1, 2, 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
Let me make it a little bit neater. So first I'll write a. So it's 1, minus 1, 1. And then minus 1, 2, 1. Minus 1, 3, 4. And then I will augment it with the identity matrix. With 1, 0, 0, 0, 1, 0, 0, 0, 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
And then minus 1, 2, 1. Minus 1, 3, 4. And then I will augment it with the identity matrix. With 1, 0, 0, 0, 1, 0, 0, 0, 1. Now, if I want to get a into reduced row echelon form, maybe I'll replace the second row. So I'll keep the first row the same for now. So let me just draw it like this. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
With 1, 0, 0, 0, 1, 0, 0, 0, 1. Now, if I want to get a into reduced row echelon form, maybe I'll replace the second row. So I'll keep the first row the same for now. So let me just draw it like this. Let me keep my first row the same. The entire first row. 1, minus 1, minus 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
So let me just draw it like this. Let me keep my first row the same. The entire first row. 1, minus 1, minus 1. It's going to be augmented with 1, 0, 0. Keep the whole first row the same. Let's replace the second row with the second row plus the first row. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
1, minus 1, minus 1. It's going to be augmented with 1, 0, 0. Keep the whole first row the same. Let's replace the second row with the second row plus the first row. So minus 1 plus 1 is 0. 2 plus minus 1 is 1. 3 plus minus 1 is 2. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
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