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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. 0 plus 1 is 0. 1 plus, oh sorry, that was a tricky one. 0 plus 1 is 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
3 plus minus 1 is 2. 0 plus 1 is 0. 1 plus, oh sorry, that was a tricky one. 0 plus 1 is 1. 1 plus 0 is 1. 0 plus 0 is 0. All I did is I added these two rows up. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
0 plus 1 is 1. 1 plus 0 is 1. 0 plus 0 is 0. All I did is I added these two rows up. Now, this third row, let me replace it. I want to get a 0 here. So let me replace the third row with the third row minus the first row. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
All I did is I added these two rows up. Now, this third row, let me replace it. I want to get a 0 here. So let me replace the third row with the third row minus the first row. So 1 minus 1 is 0. 1 minus minus 1 is 2. 4 minus minus 1 is 5. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
So let me replace the third row with the third row minus the first row. So 1 minus 1 is 0. 1 minus minus 1 is 2. 4 minus minus 1 is 5. 0 minus 1 is minus 1. 0 minus 0 is 0. And then 1 minus 0 is 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
4 minus minus 1 is 5. 0 minus 1 is minus 1. 0 minus 0 is 0. And then 1 minus 0 is 1. Just like that. Now, what do we want to do? Well, we've gotten this far. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
And then 1 minus 0 is 1. Just like that. Now, what do we want to do? Well, we've gotten this far. We want to zero out that guy and that guy. So let's keep our second row the same. Let me write it down here. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
Well, we've gotten this far. We want to zero out that guy and that guy. So let's keep our second row the same. Let me write it down here. Let's keep our second row the same. So it's 0, 1, 2. And then you augmented it with 1, 1, 0. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
Let me write it down here. Let's keep our second row the same. So it's 0, 1, 2. And then you augmented it with 1, 1, 0. Just like that. And let's replace my first row with the first row plus the second row. So 1 plus 0 is 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
And then you augmented it with 1, 1, 0. Just like that. And let's replace my first row with the first row plus the second row. So 1 plus 0 is 1. Minus 1 plus 1 is 0. That's why I did that, to get a 0 there. Minus 1 plus 2 is 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
So 1 plus 0 is 1. Minus 1 plus 1 is 0. That's why I did that, to get a 0 there. Minus 1 plus 2 is 1. 1 plus 1 is 2. 0 plus 1 is 1. 0 plus 0 is 0. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
Minus 1 plus 2 is 1. 1 plus 1 is 2. 0 plus 1 is 1. 0 plus 0 is 0. And now I also want to zero out this guy right here. So let's replace the third row with the third row minus 2 times the second row. So 0 minus 2 times 0 is 0. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
0 plus 0 is 0. And now I also want to zero out this guy right here. So let's replace the third row with the third row minus 2 times the second row. So 0 minus 2 times 0 is 0. 2 minus 2 times 1 is 0. 5 minus 2 times 2 is 5 minus 4. That's 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
So 0 minus 2 times 0 is 0. 2 minus 2 times 1 is 0. 5 minus 2 times 2 is 5 minus 4. That's 1. Minus 1 minus 2 times 1 is minus 3. 0 minus 2 times 1, that's minus 2. And then 1 minus 2 times 0 is just 1 again. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
That's 1. Minus 1 minus 2 times 1 is minus 3. 0 minus 2 times 1, that's minus 2. And then 1 minus 2 times 0 is just 1 again. All right, home stretch. Now I just want to zero out these guys right here. I just want to zero out those guys right there. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
And then 1 minus 2 times 0 is just 1 again. All right, home stretch. Now I just want to zero out these guys right here. I just want to zero out those guys right there. So let me just keep my third row the same. Let me switch colors, keep things colorful. I'm going to keep my third row the same, so it's 0, 0, 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
I just want to zero out those guys right there. So let me just keep my third row the same. Let me switch colors, keep things colorful. I'm going to keep my third row the same, so it's 0, 0, 1. I'm going to augment it with minus 3, minus 2, and 1. Now let's replace our first row with the first row minus the third row. S... | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
I'm going to keep my third row the same, so it's 0, 0, 1. I'm going to augment it with minus 3, minus 2, and 1. Now let's replace our first row with the first row minus the third row. So 1 minus 0 is 1. 0 minus 0 is 0. 1 minus 1 is 0. 2 minus minus 3, that's 5. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
So 1 minus 0 is 1. 0 minus 0 is 0. 1 minus 1 is 0. 2 minus minus 3, that's 5. 1 minus minus 2 is 3. 0 minus 1 is minus 1. Now let's replace the second row with the second row minus 2 times the third row. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
2 minus minus 3, that's 5. 1 minus minus 2 is 3. 0 minus 1 is minus 1. Now let's replace the second row with the second row minus 2 times the third row. So 0 minus 2 times 0 is 0. 1 minus 2 times 0 is 0. 2 minus 2 times 1 is 0. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
Now let's replace the second row with the second row minus 2 times the third row. So 0 minus 2 times 0 is 0. 1 minus 2 times 0 is 0. 2 minus 2 times 1 is 0. 1 minus 2 times 0 is 1. It's not 0. 2 minus 2 times 1 is 0. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
2 minus 2 times 1 is 0. 1 minus 2 times 0 is 1. It's not 0. 2 minus 2 times 1 is 0. 1 minus 2 times minus 3. That is 1 plus 2 times 3. That is 7. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
2 minus 2 times 1 is 0. 1 minus 2 times minus 3. That is 1 plus 2 times 3. That is 7. 1 minus 2 times minus 2. That's 1 plus 4, which is 5. And then 0 minus 2 times 1. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
That is 7. 1 minus 2 times minus 2. That's 1 plus 4, which is 5. And then 0 minus 2 times 1. So that's minus 2. And just like that, we've gotten the A part of our augmented matrix into reduced row echelon form. This is the reduced row echelon form of A. | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
And then 0 minus 2 times 1. So that's minus 2. And just like that, we've gotten the A part of our augmented matrix into reduced row echelon form. This is the reduced row echelon form of A. And when you apply those exact same transformations, because if you think about it, that series of matrix products that got you fro... | Example of finding matrix inverse Matrix transformations Linear Algebra Khan Academy.mp3 |
So if I have some matrix, let's just call it B. If my matrix B looks like this, if its entries are A, B, C, D, we defined the determinant of B, which could also be written as B with these lines around it, which could also be written as the entries of the matrix with those lines around it. A, B, C, D. And I don't want y... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
This is the matrix when you have the brackets. This is the determinant of the matrix when you just have these straight lines. And this, by definition, was equal to A, D minus B, C. And you saw in the last video, or maybe you saw in the last video, what the motivation for this came from. When we figured out the inverse ... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
When we figured out the inverse of B, we determined that it was equal to the inverse of B was equal to 1 over A, D minus B, C times another matrix, which was essentially these two entries swapped. So you got a D and an A. And then these two entries made negative. So minus C and minus B. This was the inverse of B. And w... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
So minus C and minus B. This was the inverse of B. And we said, well, when is this defined? This is defined as long as this character right here does not equal 0. So we said, hey, this looks pretty important. Let's call this thing right there the determinant. Let's call this right here the determinant. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
This is defined as long as this character right here does not equal 0. So we said, hey, this looks pretty important. Let's call this thing right there the determinant. Let's call this right here the determinant. And then we could say that B is invertible if and only if the determinant of B does not equal 0. Because if ... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Let's call this right here the determinant. And then we could say that B is invertible if and only if the determinant of B does not equal 0. Because if it equals 0, then this formula for your inverse won't be well-defined. We just got this from our technique of creating an augmented matrix, whatnot. But the big takeawa... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
We just got this from our technique of creating an augmented matrix, whatnot. But the big takeaway is we defined this notion of a determinant for a 2 by 2 matrix. Now, the next question is, well, that's just a 2 by 2. Everything we do in linear algebra, we like to generalize it to higher numbers of rows and columns. So... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Everything we do in linear algebra, we like to generalize it to higher numbers of rows and columns. So the next step, at least, let's just do baby steps. Let's start with a 3 by 3. Let's define what its determinant is. So let me construct a 3 by 3 matrix here. Let's say my matrix A is equal to, let me just write its en... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Let's define what its determinant is. So let me construct a 3 by 3 matrix here. Let's say my matrix A is equal to, let me just write its entries, first row, first column, first row, second column, first row, third column. Then you have A21, A22, A23, then you have A31, third row, first column, A32, and then A33. That i... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Then you have A21, A22, A23, then you have A31, third row, first column, A32, and then A33. That is a 3 by 3 matrix. Clearly, three rows and three columns. This is 3 by 3. I am going to define the determinant of A. So this is a definition. I'm going to define the determinant of this 3 by 3 matrix A as being equal to, a... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
This is 3 by 3. I am going to define the determinant of A. So this is a definition. I'm going to define the determinant of this 3 by 3 matrix A as being equal to, and this is a little bit convoluted. But you'll get the hang of it eventually. In the next several videos, we're just going to do a ton of determinants, so i... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
I'm going to define the determinant of this 3 by 3 matrix A as being equal to, and this is a little bit convoluted. But you'll get the hang of it eventually. In the next several videos, we're just going to do a ton of determinants, so it just becomes a bit of second nature to you. It's a little computationally intensiv... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
It's a little computationally intensive sometimes. But it equals this first row, it equals A11 times the determinant of the matrix you get if you get rid of this guy's column and row. So if you get rid of this guy's column and row, you're left with this matrix here. So times the determinant of the matrix A22, A23, A32,... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
So times the determinant of the matrix A22, A23, A32, and then A33, just like that. So that's our first entry, and that's a plus this. And then I said it's a plus this because the next entry is going to be a minus. You have a minus this guy right here. So then you're going to have minus A12 times the matrix you get if ... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
You have a minus this guy right here. So then you're going to have minus A12 times the matrix you get if you eliminate his column and his row. So times, you're going to get these entries right there. So A21, A21, A23, A23, A31, A31, and then you have A33. We're not quite done. You could probably guess what the next one... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
So A21, A21, A23, A23, A31, A31, and then you have A33. We're not quite done. You could probably guess what the next one's going to be. And you're going to have a plus, let me switch to a better color, plus this guy, plus A13 times the determinant of its, I guess you could call it its sub-matrix. We'll call it that for... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
And you're going to have a plus, let me switch to a better color, plus this guy, plus A13 times the determinant of its, I guess you could call it its sub-matrix. We'll call it that for now. So this matrix right here. So A21, A21, A22, A31, A32. This is our definition of the determinant of a 3 by 3 matrix. And the motiv... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
So A21, A21, A22, A31, A32. This is our definition of the determinant of a 3 by 3 matrix. And the motivation is because when you take the determinant of a 3 by 3, it turns out, I haven't shown it to you yet, that the property is the same. That if the determinant of this is 0, you will not be able to find an inverse. An... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
That if the determinant of this is 0, you will not be able to find an inverse. And when I define the determinant in this way, if the determinant does not equal 0, you will be able to find an inverse. So that's where this came from. And I haven't shown you that yet. And I might not show you, because it's super computati... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
And I haven't shown you that yet. And I might not show you, because it's super computational. It'll take a long time. It'll be very hairy, and I'll make careless mistakes. But the motivation comes from the exact same place as the 2 by 2 version. But I think what you probably want to see right now is at least just this ... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
It'll be very hairy, and I'll make careless mistakes. But the motivation comes from the exact same place as the 2 by 2 version. But I think what you probably want to see right now is at least just this thing applied to an actual matrix. Because this looks all abstract right now. But if you do it with an actual matrix, ... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Because this looks all abstract right now. But if you do it with an actual matrix, you'll actually see it's not too bad. So let's leave the definition up there. And let's say that I have the matrix 1, 2, 4, 2, 2, minus 1, 3, and 4, 0, 1. So by our definition of a determinant, the determinant of this guy right here, the... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
And let's say that I have the matrix 1, 2, 4, 2, 2, minus 1, 3, and 4, 0, 1. So by our definition of a determinant, the determinant of this guy right here, the determinant of this guy, so let's say I call that matrix, let's call that c. c is equal to that. So if we want to figure out the determinant of c, the determina... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
So we have a minus 1, minus 1, we have a, sorry, we have a minus 1, got to be careful, we have a 3, we have a 0, and we have a 1, just like that. Notice I got rid of this guy's column and this guy's row, and I was just left with minus 1, 3, 0, 1. Minus 1, 3, 0, and 1. Next, I take this guy. And this is the trick. You h... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Next, I take this guy. And this is the trick. You have to alternate signs. If you start with a positive here, this next one's going to be a minus. So you're going to have minus 2 times the sub-matrix, we can call it, if we get rid of this guy's column and this guy's row. So 2, 3, 4, 1. I just blanked this out. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
If you start with a positive here, this next one's going to be a minus. So you're going to have minus 2 times the sub-matrix, we can call it, if we get rid of this guy's column and this guy's row. So 2, 3, 4, 1. I just blanked this out. If I could videotape my finger, I would cover my finger over this column right here... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
I just blanked this out. If I could videotape my finger, I would cover my finger over this column right here and over that row, and you'd just see a 2, a 3, a 4, and a 1. And that's what I put right there. And then finally, we'll have plus, we went plus, minus, plus, so finally we'll have plus 4 times the determinant o... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
And then finally, we'll have plus, we went plus, minus, plus, so finally we'll have plus 4 times the determinant of the sub-matrix, if you get rid of that row and that column. So 2, minus 1, 4, 0. 2, minus 1, 4, and 0. Now these are pretty straightforward. These are not too bad to compute. Let's actually do it. So this... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Now these are pretty straightforward. These are not too bad to compute. Let's actually do it. So this is going to be equal to 1 times what? Minus 1 times 1. Let me just write it out. Minus 1 times 1 minus 0 times 3. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
So this is going to be equal to 1 times what? Minus 1 times 1. Let me just write it out. Minus 1 times 1 minus 0 times 3. This just comes from the definition of a 2 by 2 determinant. We've already defined that. And then we're going to have a minus 2 times 2 times 1 minus 4 times 3. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Minus 1 times 1 minus 0 times 3. This just comes from the definition of a 2 by 2 determinant. We've already defined that. And then we're going to have a minus 2 times 2 times 1 minus 4 times 3. And then finally, we're going to have a plus 4 times 2 times 0 minus minus 1 minus 1 times 4. Minus minus 1 times 4. I wrote i... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
And then we're going to have a minus 2 times 2 times 1 minus 4 times 3. And then finally, we're going to have a plus 4 times 2 times 0 minus minus 1 minus 1 times 4. Minus minus 1 times 4. I wrote it all out. So you can see that these are just, this thing right here is just this thing right here. And then you have the ... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
I wrote it all out. So you can see that these are just, this thing right here is just this thing right here. And then you have the 4 out front. This thing right here was just this thing right here. It's just the determinant of the 2 by 2 sub-matrix for each of these guys. And if we compute this, this is equal to minus ... | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
This thing right here was just this thing right here. It's just the determinant of the 2 by 2 sub-matrix for each of these guys. And if we compute this, this is equal to minus 1 times 1 is minus 1 minus 0. That's 0. So this is a minus 1 times 1. So that's a minus 1. And then we get, what is this equal to? | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
That's 0. So this is a minus 1 times 1. So that's a minus 1. And then we get, what is this equal to? This right here is 12. So you get 2 minus 12. You get 2 times 1 minus 4 times 3. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
And then we get, what is this equal to? This right here is 12. So you get 2 minus 12. You get 2 times 1 minus 4 times 3. So it's minus 10. So that is equal to minus 10. And you have a minus 10 times a minus 2. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
You get 2 times 1 minus 4 times 3. So it's minus 10. So that is equal to minus 10. And you have a minus 10 times a minus 2. So that becomes a plus 20. Minus 2 times minus 10. And then finally in the green, we have 2 times 0. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
And you have a minus 10 times a minus 2. So that becomes a plus 20. Minus 2 times minus 10. And then finally in the green, we have 2 times 0. That's just a 0. And then you have minus 1 times 4, which is minus 4. Then you have a minus sign here. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
And then finally in the green, we have 2 times 0. That's just a 0. And then you have minus 1 times 4, which is minus 4. Then you have a minus sign here. So it's plus 4. And then so this all becomes a plus 4. Plus 4 times 4 is 16. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Then you have a minus sign here. So it's plus 4. And then so this all becomes a plus 4. Plus 4 times 4 is 16. So plus 16. And what do we get when we add this up? We get 20 plus 16 minus 1. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Plus 4 times 4 is 16. So plus 16. And what do we get when we add this up? We get 20 plus 16 minus 1. It is equal to 35. We're done. We found the determinant of our 3 by 3 matrix. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
We get 20 plus 16 minus 1. It is equal to 35. We're done. We found the determinant of our 3 by 3 matrix. Not too bad. Right there. So that is equal to the determinant of c. So the fact that this isn't 0 tells you that c is invertible. | 3 x 3 determinant Matrix transformations Linear Algebra Khan Academy.mp3 |
Let's say I had the vectors, let's say I had the set of vectors, I don't want to do it that thick. Let's say one of the vectors is the vector 2, 3, and then the other vector is the vector 4, 6. And I just want to answer the question, what is the span of these vectors? Let's assume that these are position vectors. What ... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's assume that these are position vectors. What are all of the vectors that these two vectors can represent? Well, if you just look at it, and remember the span is just all of the vectors that can be represented by linear combinations of these. So it's the set of all the vectors, and if I have some constant times 2 ... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
So it's the set of all the vectors, and if I have some constant times 2 times that vector, plus some other constant times this vector, it's all the possibilities that I can represent when I just put a bunch of different real numbers for c1 and c2. Now the first thing you might realize is that, look, this vector 2, this... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
I could just rewrite it as c1 times the vector 2, 3, plus c2 times the vector, and here I'm going to write, instead of writing the vector 4, 6, I'm going to write 2 times the vector 2, 3. Because this vector is just a multiple of that vector, so I could write c2 times 2 times 2, 3. I think you see that this is equivale... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
2 times 2 is 4, 2 times 3 is 6. Well then we can simplify this a little bit. We can rewrite this as just c1 plus 2c2, all of that, times 2, 3, times our vector 2, 3. And this is just some arbitrary constant. It's some arbitrary constant plus 2 times some other arbitrary constant. So we could just call this c3 times my ... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
And this is just some arbitrary constant. It's some arbitrary constant plus 2 times some other arbitrary constant. So we could just call this c3 times my vector 2, 3. In this situation, even though we started with two vectors, and I said, well, you know, the span of these two vectors is equal to all of the vectors that... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
In this situation, even though we started with two vectors, and I said, well, you know, the span of these two vectors is equal to all of the vectors that can be constructed with some linear combination of these, any linear combination of these, if I just use this substitution right here, can be reduced to just a scalar... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
But the fact is, is that instead of talking about linear combinations of two vectors, I can reduce this to just a scalar combination of one vector. And we've seen in R2 a scalar combination of one vector, especially if they're position vectors. For example, this vector 2, 3, it's 2, 3, it looks like this. All the scala... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
All the scalar combinations of that vector are just going to lie along this line. So 2, 3, it's going to be right there. They're all just going to lie along that line right there. So 5 and then 4, along this line, going in both directions forever. And if I take a negative values of 2, 3, I'm going to go down here. If I... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
So 5 and then 4, along this line, going in both directions forever. And if I take a negative values of 2, 3, I'm going to go down here. If I take positive values, I'm going to go here. If I get really large positive values, I'm going to go up here. But I can just represent the vectors, and when you put them in standard... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
If I get really large positive values, I'm going to go up here. But I can just represent the vectors, and when you put them in standard form, their arrows essentially would trace out this line. So you could say that the span of my set of vectors, let me put it over here, the span of the vectors, let me do it this way, ... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
Even though we have two vectors, they're essentially collinear, they're multiples of each other. I mean, 4, 6 is, if this is 2, 3, 4, 6 is just this right here. 4, 6, it's just that longer one right there. They're collinear. These two things are collinear. Now, so in this case, when we have two collinear vectors in R2,... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
They're collinear. These two things are collinear. Now, so in this case, when we have two collinear vectors in R2, they essentially, their span just reduces to that line. You can't represent some vector like, you can't represent, let me do a new color, you can't represent this vector right there with some combination o... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
You can't represent some vector like, you can't represent, let me do a new color, you can't represent this vector right there with some combination of those two vectors. There's no way to kind of break out of this line. So there's no way that you can represent everything in R2. So the span is just that line there. Now,... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
So the span is just that line there. Now, a related idea to this, notice, you had two vectors, but it kind of reduced to one vector when you took its linear combinations. The related idea here is that we call this set, we call it linearly dependent. Let me write that down. Linearly dependent. This is a linearly depende... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let me write that down. Linearly dependent. This is a linearly dependent set. And linearly dependent just means that one of the vectors in the set can be represented by some combination of the other vectors in the set. And a way to think about it is, whichever vector you pick that can be represented by the others, it's... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
And linearly dependent just means that one of the vectors in the set can be represented by some combination of the other vectors in the set. And a way to think about it is, whichever vector you pick that can be represented by the others, it's not adding any new directionality or any new information, right? In this case... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
And when you throw this 4, 6 on there, you're going in the same direction, just scaled up. So it's not adding us, it's not giving us any new dimension, letting us break out of this line, right? And you can imagine in 3-space, if you have one vector that looks like this, and another vector that looks like this, two vect... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
In order to define R3, a third vector in that set can't be coplanar with those two. If this third vector is coplanar with these, it's not adding any more directionality. So this set of three vectors will also be linearly dependent. And another way to think about it is that these two purple vectors span this plane, span... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
And another way to think about it is that these two purple vectors span this plane, span the plane that they define, essentially, right? Anything in this plane going in any direction can be, any vector in this plane, when we say span it, that means that any vector can be represented by a linear combination of this vect... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
This one can be represented by a sum of that one and that one, because this one and this one span this plane. In order for the span of these three vectors to kind of get more dimensionality or start representing R3, the third vector will have to break out of that plane. And if a vector is breaking out of that plane, th... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
Or it's outside, it can't be represented by a linear combination of this one and this one. So if you had a vector of this one, this one, and this one, and just those three, none of these other things that I drew, that would be linearly independent. Let me draw a couple more examples for you. That one might have been a ... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
That one might have been a little too abstract. For example, if I had the vectors 2, 3, and I have the vector 7, 2, and I have the vector 9, 5, and I were to ask you, are these linearly dependent or independent? So at first you say, well, you know, it's not trivial. Let's see, this isn't a scalar multiple of that. That... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's see, this isn't a scalar multiple of that. That doesn't look like a scalar multiple of either of the other two. Maybe they're linearly independent. But then if you kind of inspect them, you kind of see that if we call this v1, vector 1 plus vector 2 is equal to vector 3. So vector 3 is a linear combination of the... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
But then if you kind of inspect them, you kind of see that if we call this v1, vector 1 plus vector 2 is equal to vector 3. So vector 3 is a linear combination of these other two vectors. So this is a linearly dependent set. And if we were to show it, draw it in kind of 2 space, it's just a general idea that or let me ... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
And if we were to show it, draw it in kind of 2 space, it's just a general idea that or let me see, let me draw it in R2. It's a general idea that if you have three two-dimensional vectors, one of them is going to be redundant. Well, one of them definitely will be redundant. For example, if we do 2, 3, if we do the vec... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
For example, if we do 2, 3, if we do the vector 2, 3, that's the first one right there, draw it in our standard position, and I draw the vector 7, 2, I could show you that any point in R2 can be represented by some linear combination of these two vectors. We could even do kind of a graphical representation. But I've do... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
So I could write that the span of v1 and v2 is equal to R2. That means that every vector, every position here can be represented by some linear combination of these two guys. Now, the vector 9, 5, it's right there, it is in R2. Clearly, I just graphed it on this plane. It's in our two-dimensional real number space, or ... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
Clearly, I just graphed it on this plane. It's in our two-dimensional real number space, or I guess we could call it a space, or in our set R2, it's there. It's right there. So we just said that anything in R2 can be represented by a linear combination of those two guys. So clearly, this is in R2, so it can be represen... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
So we just said that anything in R2 can be represented by a linear combination of those two guys. So clearly, this is in R2, so it can be represented as a linear combination. So hopefully you're starting to see the relationship between span and linear independence or linear dependence. Let me do another example. Let's ... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let me do another example. Let's say I have the vectors. Let me do a new color. Let's say I have the vector, and this will be a little bit obvious, 7, 0. So that's my v1. And then I have my second vector, which is 0, minus 1. That's v2. | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's say I have the vector, and this will be a little bit obvious, 7, 0. So that's my v1. And then I have my second vector, which is 0, minus 1. That's v2. Now, is this set linearly independent? Well, can I represent either of these as a combination of the other? And really, when I say as a combination, you would have... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
That's v2. Now, is this set linearly independent? Well, can I represent either of these as a combination of the other? And really, when I say as a combination, you would have to scale up one to get the other because there's only two vectors here. Now, trying to add up to this vector, the only thing I have to deal with ... | Introduction to linear independence Vectors and spaces Linear Algebra Khan Academy.mp3 |
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