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I think you can see immediately that if you were to add this to this, you would get this right here, right? Because you would have this coefficient and this coefficient on that. If you added them up, you would get minus x1 plus y1. This guy and this guy add up to this guy. And then if I were to do this guy and this guy add up to that guy. And let me do another one. And finally, that term plus that term add up to that term. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
This guy and this guy add up to this guy. And then if I were to do this guy and this guy add up to that guy. And let me do another one. And finally, that term plus that term add up to that term. So you immediately see that the determinant, or hopefully you immediately see that the determinant of x plus the determinant of y is equal to the determinant of z. So we did it for the 2 by 2 case. We just did it for the 3 by 3 case. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
And finally, that term plus that term add up to that term. So you immediately see that the determinant, or hopefully you immediately see that the determinant of x plus the determinant of y is equal to the determinant of z. So we did it for the 2 by 2 case. We just did it for the 3 by 3 case. Might as well do it for the n by n case so we know that it works. But the argument is identical to this 3 by 3 case. And it's good to keep in your mind, because 3 by 3 is easy to visualize. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
We just did it for the 3 by 3 case. Might as well do it for the n by n case so we know that it works. But the argument is identical to this 3 by 3 case. And it's good to keep in your mind, because 3 by 3 is easy to visualize. n by n is sometimes a little bit abstract. So let me define, let me redefine my matrices again. I'm just going to do the same thing over again. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
And it's good to keep in your mind, because 3 by 3 is easy to visualize. n by n is sometimes a little bit abstract. So let me define, let me redefine my matrices again. I'm just going to do the same thing over again. So I'm going to have a matrix x, but it's an n by n matrix. So let me write it this way. Let's say it is a11, a12, all the way to a1n. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
I'm just going to do the same thing over again. So I'm going to have a matrix x, but it's an n by n matrix. So let me write it this way. Let's say it is a11, a12, all the way to a1n. And there's some row here, let's say that there's some row here on row i. Let's call this row i right here. And here it is, has the terms x1, x2, all the way to xn. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
Let's say it is a11, a12, all the way to a1n. And there's some row here, let's say that there's some row here on row i. Let's call this row i right here. And here it is, has the terms x1, x2, all the way to xn. But then everything else is just the regular a's. So then you have a, let me make this as a21, all the way to a2n, and then if you went all the way down here, you would have a n1, and you'd go all the way to a nn. So essentially you could imagine our standard matrix where everything is defined in a, but I replaced row i with certain numbers that are maybe a little different. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
And here it is, has the terms x1, x2, all the way to xn. But then everything else is just the regular a's. So then you have a, let me make this as a21, all the way to a2n, and then if you went all the way down here, you would have a n1, and you'd go all the way to a nn. So essentially you could imagine our standard matrix where everything is defined in a, but I replaced row i with certain numbers that are maybe a little different. And I think you'll see where I'm going. Now let me define my other matrix. Let me define matrix y. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
So essentially you could imagine our standard matrix where everything is defined in a, but I replaced row i with certain numbers that are maybe a little different. And I think you'll see where I'm going. Now let me define my other matrix. Let me define matrix y. Let me define matrix y to be essentially the same thing. This is a11, it's the same a11. This is a12, all the way to a1n. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
Let me define matrix y. Let me define matrix y to be essentially the same thing. This is a11, it's the same a11. This is a12, all the way to a1n. This is a21, we could go all the way to a2n. And then on row i, the same row, this is n by n, this is the same n by n. If this was 10 by 10, this is 10 by 10. If this is row 7, then this is row 7. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
This is a12, all the way to a1n. This is a21, we could go all the way to a2n. And then on row i, the same row, this is n by n, this is the same n by n. If this was 10 by 10, this is 10 by 10. If this is row 7, then this is row 7. It has different terms. It's identical to matrix x, except for row i. In row i it is y1, y2, all the way to yn. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
If this is row 7, then this is row 7. It has different terms. It's identical to matrix x, except for row i. In row i it is y1, y2, all the way to yn. And if you keep going down, of course you have a n1, all the way to a nn. Fair enough. Now let's say we have a third matrix. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
In row i it is y1, y2, all the way to yn. And if you keep going down, of course you have a n1, all the way to a nn. Fair enough. Now let's say we have a third matrix. Let's have a third matrix. Let me draw it right here. So you have z. z is equal to, I think you could imagine where this is going. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
Now let's say we have a third matrix. Let's have a third matrix. Let me draw it right here. So you have z. z is equal to, I think you could imagine where this is going. z is identical to these two guys, except for row i. So let me write that out. So z looks like this. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
So you have z. z is equal to, I think you could imagine where this is going. z is identical to these two guys, except for row i. So let me write that out. So z looks like this. You have a11, a12, a12, all the way to a1n. And then you go down. And then row i happens to be the sum of the row i of matrix x and matrix y. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
So z looks like this. You have a11, a12, a12, all the way to a1n. And then you go down. And then row i happens to be the sum of the row i of matrix x and matrix y. So it is x1 plus y1, x2 plus y2, all the way to xn plus yn. And then you keep going down. Everything else is identical. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
And then row i happens to be the sum of the row i of matrix x and matrix y. So it is x1 plus y1, x2 plus y2, all the way to xn plus yn. And then you keep going down. Everything else is identical. a n1, all the way to a nn. So all of these matrices are identical, except for row x has a different row i than matrix y does. And row z is identical everywhere, except its row i is the sum of this row i and that row i. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
Everything else is identical. a n1, all the way to a nn. So all of these matrices are identical, except for row x has a different row i than matrix y does. And row z is identical everywhere, except its row i is the sum of this row i and that row i. So it's a very particular case. But we can figure out their determinants. So what are the determinants? | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
And row z is identical everywhere, except its row i is the sum of this row i and that row i. So it's a very particular case. But we can figure out their determinants. So what are the determinants? The determinant of x, the determinant of matrix x. Hopefully you're maybe a bit comfortable with writing sigma notation. We did this in the last matrix. We can go down this row. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
So what are the determinants? The determinant of x, the determinant of matrix x. Hopefully you're maybe a bit comfortable with writing sigma notation. We did this in the last matrix. We can go down this row. We can go down this row right there. And for each of these guys, we can say that this, so the determinant is going to be equal to the sum. Let's say we start from j is equal to 1. j is going to be the column. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
We can go down this row. We can go down this row right there. And for each of these guys, we can say that this, so the determinant is going to be equal to the sum. Let's say we start from j is equal to 1. j is going to be the column. So we're going to take the sum of each of these terms. From j is equal to 1 to n. And then remember our checkerboard pattern. So we don't know if this is a positive or negative. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
Let's say we start from j is equal to 1. j is going to be the column. So we're going to take the sum of each of these terms. From j is equal to 1 to n. And then remember our checkerboard pattern. So we don't know if this is a positive or negative. But we can figure it out by taking negative 1 to the i plus j. Remember, this is the i-th row that we're talking about. Times xj. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
So we don't know if this is a positive or negative. But we can figure it out by taking negative 1 to the i plus j. Remember, this is the i-th row that we're talking about. Times xj. xj is the coefficient. x sub j. Times the sub matrix for x sub j. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
Times xj. xj is the coefficient. x sub j. Times the sub matrix for x sub j. So if you get rid of this guy's row and this guy's column, what is it going to be? We could say that that's the same thing as the sub matrix. If we called this guy, let me write it this way. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
Times the sub matrix for x sub j. So if you get rid of this guy's row and this guy's column, what is it going to be? We could say that that's the same thing as the sub matrix. If we called this guy, let me write it this way. If we got rid of this guy's row and this guy's column, if we had just our traditional matrix where this wasn't replaced, if we just had an ai1 here, ai2, its sub matrix would be the same thing. Because we're crossing out this row and this column. So it would be all of these guys and all of these guys down here. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
If we called this guy, let me write it this way. If we got rid of this guy's row and this guy's column, if we had just our traditional matrix where this wasn't replaced, if we just had an ai1 here, ai2, its sub matrix would be the same thing. Because we're crossing out this row and this column. So it would be all of these guys and all of these guys down here. So it would be the sub matrix. This is a n minus 1 by n minus 1 matrix. It would be the sub matrix for aij. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
So it would be all of these guys and all of these guys down here. So it would be the sub matrix. This is a n minus 1 by n minus 1 matrix. It would be the sub matrix for aij. And then that's for the first term. Sorry, the determinant. Don't want to lose the determinant there. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
It would be the sub matrix for aij. And then that's for the first term. Sorry, the determinant. Don't want to lose the determinant there. Times the determinant of the sub matrix aij. And so that's for the first term. And then you're going to add it to the second term. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
Don't want to lose the determinant there. Times the determinant of the sub matrix aij. And so that's for the first term. And then you're going to add it to the second term. And then you're just going to keep doing that. And that's what this sigma notation is. That's the determinant of x. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
And then you're going to add it to the second term. And then you're just going to keep doing that. And that's what this sigma notation is. That's the determinant of x. Now, what's the determinant of y? The determinant of y is equal to the sum. We could do the same thing. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
That's the determinant of x. Now, what's the determinant of y? The determinant of y is equal to the sum. We could do the same thing. j is equal to 1 to n of negative 1 to the i plus j. Each of them, we're going to go along this row right here. The ith row. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
We could do the same thing. j is equal to 1 to n of negative 1 to the i plus j. Each of them, we're going to go along this row right here. The ith row. So we're going to have y sub j. We're going to start with y sub 1, then plus y sub 2. Times the determinant of its sub matrix. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
The ith row. So we're going to have y sub j. We're going to start with y sub 1, then plus y sub 2. Times the determinant of its sub matrix. Which is the same as the determinant of this sub matrix. When you get rid of that row and that column for each of these guys, everything else on the matrix is the same. So aij. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
Times the determinant of its sub matrix. Which is the same as the determinant of this sub matrix. When you get rid of that row and that column for each of these guys, everything else on the matrix is the same. So aij. The matrix of aij. Now, what is the determinant of z? I'm pretty sure you know exactly where this is going. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
So aij. The matrix of aij. Now, what is the determinant of z? I'm pretty sure you know exactly where this is going. The determinant, this should be a capital Y right there. The determinant of z is equal to the sum from j is equal to 1 to n of negative 1 to the i plus j. We're going along this row. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
I'm pretty sure you know exactly where this is going. The determinant, this should be a capital Y right there. The determinant of z is equal to the sum from j is equal to 1 to n of negative 1 to the i plus j. We're going along this row. But now the coefficients are xj. That's what we're indexing along. xj plus yj. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
We're going along this row. But now the coefficients are xj. That's what we're indexing along. xj plus yj. And then times its sub matrix, which is the same as these sub matrices. So aij. aij. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
xj plus yj. And then times its sub matrix, which is the same as these sub matrices. So aij. aij. Which you might immediately see is the sum of these two things. If I, for every j, I just summed these two things up, you're having two coefficients. You could have this coefficient and that coefficient on your aij term. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
aij. Which you might immediately see is the sum of these two things. If I, for every j, I just summed these two things up, you're having two coefficients. You could have this coefficient and that coefficient on your aij term. And then when you add them up, you can factor this guy out and you'll get this right here. You will get this right here. So you get the determinant of x plus the determinant of y is equal to the determinant of z. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
You could have this coefficient and that coefficient on your aij term. And then when you add them up, you can factor this guy out and you'll get this right here. You will get this right here. So you get the determinant of x plus the determinant of y is equal to the determinant of z. So hopefully that shows you the general case. But I want to make it very clear. This is just for a very particular scenario where three matrices are identical except for on one row. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
So you get the determinant of x plus the determinant of y is equal to the determinant of z. So hopefully that shows you the general case. But I want to make it very clear. This is just for a very particular scenario where three matrices are identical except for on one row. And one of the matrices on that special row just happens to be the sum of the other two matrices for that special row. And everything else is identical. That's the only time where the determinant of z, not the only time, but that's the only time we can make the general statement where the determinant of z is equal to the determinant of x plus the determinant of y. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
This is just for a very particular scenario where three matrices are identical except for on one row. And one of the matrices on that special row just happens to be the sum of the other two matrices for that special row. And everything else is identical. That's the only time where the determinant of z, not the only time, but that's the only time we can make the general statement where the determinant of z is equal to the determinant of x plus the determinant of y. It's not the case, let me write what is not the case. So not the case that if, let's say, z is equal to x plus y, it is not the case that the determinant of z is necessarily equal to the determinant of x plus the determinant of y. You cannot assume this. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
That's the only time where the determinant of z, not the only time, but that's the only time we can make the general statement where the determinant of z is equal to the determinant of x plus the determinant of y. It's not the case, let me write what is not the case. So not the case that if, let's say, z is equal to x plus y, it is not the case that the determinant of z is necessarily equal to the determinant of x plus the determinant of y. You cannot assume this. Determinant operations are not linear on matrix addition. They're linear only on particular rows getting added. Anyway, hopefully you found that vaguely clear. | Determinant when row is added Matrix transformations Linear Algebra Khan Academy.mp3 |
And that says, well, that means any value lambda that satisfies this equation for v is not a non-zero vector. We just did a little bit of vector algebra up here to come up with that. You can review that video if you like. And then we determined, look, the only way that this is going to have a non-zero solution is if this matrix has a non-trivial null space and only non-invertible matrices have a non-trivial null space, or only matrices that have a determinant of 0 have non-trivial null spaces. So you do that, you got your characteristic polynomial, and we were able to solve it. And we got our eigenvalues where lambda is equal to 3 and lambda is equal to minus 3. So now let's do what I consider the more interesting part, is actually find out the eigenvectors, or the eigenspaces. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And then we determined, look, the only way that this is going to have a non-zero solution is if this matrix has a non-trivial null space and only non-invertible matrices have a non-trivial null space, or only matrices that have a determinant of 0 have non-trivial null spaces. So you do that, you got your characteristic polynomial, and we were able to solve it. And we got our eigenvalues where lambda is equal to 3 and lambda is equal to minus 3. So now let's do what I consider the more interesting part, is actually find out the eigenvectors, or the eigenspaces. So we can go back to this equation. For any eigenvalue, this must be true, but this is easier to work with. And so this matrix right here times your eigenvector must be equal to 0 for any given eigenvalue. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So now let's do what I consider the more interesting part, is actually find out the eigenvectors, or the eigenspaces. So we can go back to this equation. For any eigenvalue, this must be true, but this is easier to work with. And so this matrix right here times your eigenvector must be equal to 0 for any given eigenvalue. This matrix right here, I just copied and pasted from above. I marked it up with the rule of Cyrus, but you could ignore those lines. It's just this matrix right here for any lambda. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And so this matrix right here times your eigenvector must be equal to 0 for any given eigenvalue. This matrix right here, I just copied and pasted from above. I marked it up with the rule of Cyrus, but you could ignore those lines. It's just this matrix right here for any lambda. Lambda times the identity matrix minus a ends up being this. So let's take this matrix for each of our lambdas and then solve for our eigenvectors, or our eigenspaces. So let's take the case of lambda is equal to 3 first. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
It's just this matrix right here for any lambda. Lambda times the identity matrix minus a ends up being this. So let's take this matrix for each of our lambdas and then solve for our eigenvectors, or our eigenspaces. So let's take the case of lambda is equal to 3 first. So if lambda is equal to 3, this matrix becomes lambda plus 1 is 4, lambda minus 2 is 1, lambda minus 2 is 1. And then all of the other terms stay the same. Minus 2, minus 2, minus 2, 1, minus 2, and 1. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So let's take the case of lambda is equal to 3 first. So if lambda is equal to 3, this matrix becomes lambda plus 1 is 4, lambda minus 2 is 1, lambda minus 2 is 1. And then all of the other terms stay the same. Minus 2, minus 2, minus 2, 1, minus 2, and 1. And then this times that vector v, or our eigenvector v, is equal to 0. Or we could say that the eigenspace for the eigenvalue 3 is the null space of this matrix, which is not this matrix. It's lambda times identity minus a. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Minus 2, minus 2, minus 2, 1, minus 2, and 1. And then this times that vector v, or our eigenvector v, is equal to 0. Or we could say that the eigenspace for the eigenvalue 3 is the null space of this matrix, which is not this matrix. It's lambda times identity minus a. So the null space of this matrix is the eigenspace. So all of the values that satisfy this make up the eigenvectors of the eigenspace of lambda is equal to 3. So let's just solve for this. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
It's lambda times identity minus a. So the null space of this matrix is the eigenspace. So all of the values that satisfy this make up the eigenvectors of the eigenspace of lambda is equal to 3. So let's just solve for this. So the null space of this guy, we could just put it in reduced row echelon form. The null space of this guy is the same thing as the null space of this guy in reduced row echelon form. So let's put it in reduced row echelon form. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So let's just solve for this. So the null space of this guy, we could just put it in reduced row echelon form. The null space of this guy is the same thing as the null space of this guy in reduced row echelon form. So let's put it in reduced row echelon form. So the first thing I want to do, let me just do it down here, so let me, I'll keep my first row the same for now. 4, minus 2, minus 2. And let me replace my second row with my second row times 2 plus my first row. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So let's put it in reduced row echelon form. So the first thing I want to do, let me just do it down here, so let me, I'll keep my first row the same for now. 4, minus 2, minus 2. And let me replace my second row with my second row times 2 plus my first row. So minus 2 times 2 plus 1 is 0. 1 times 2 plus minus 2 is 0. 1 times 2 plus minus 2 is 0. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And let me replace my second row with my second row times 2 plus my first row. So minus 2 times 2 plus 1 is 0. 1 times 2 plus minus 2 is 0. 1 times 2 plus minus 2 is 0. This row is the same as this row, so I'm going to do the same thing. Minus 2 times 2 plus 4 is 0. 1 times 2 plus 2 is 0. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
1 times 2 plus minus 2 is 0. This row is the same as this row, so I'm going to do the same thing. Minus 2 times 2 plus 4 is 0. 1 times 2 plus 2 is 0. And then 1 times 2 plus minus 2 is 0. So the solutions to this equation are the same as the solutions to this equation. Let me write it like this. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
1 times 2 plus 2 is 0. And then 1 times 2 plus minus 2 is 0. So the solutions to this equation are the same as the solutions to this equation. Let me write it like this. Instead of just writing the vector v, let me write it out. So v1, v2, v3 are going to be equal to the 0 vector, 0, 0. Just rewriting it slightly different. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Let me write it like this. Instead of just writing the vector v, let me write it out. So v1, v2, v3 are going to be equal to the 0 vector, 0, 0. Just rewriting it slightly different. And so these two rows, or these two equations, give us no information. The only one is this row up here, which tells us that 4 times v1 minus 2 times v2, actually this wasn't complete reduced row echelon form, but close enough. It's easy for us to work with. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Just rewriting it slightly different. And so these two rows, or these two equations, give us no information. The only one is this row up here, which tells us that 4 times v1 minus 2 times v2, actually this wasn't complete reduced row echelon form, but close enough. It's easy for us to work with. 4 times v1 minus 2 times v2 minus 2 times v3 is equal to 0, and let's just divide by 4. I could have just divided by 4 here, which might have made me skip a step. If you divide by 4, you get v1 minus 1 half v2 minus 1 half v3 is equal to 0. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
It's easy for us to work with. 4 times v1 minus 2 times v2 minus 2 times v3 is equal to 0, and let's just divide by 4. I could have just divided by 4 here, which might have made me skip a step. If you divide by 4, you get v1 minus 1 half v2 minus 1 half v3 is equal to 0. Or v1 is equal to 1 half v2 plus 1 half v3. I just added these guys to both sides of the equation. Or we could say, if we say that v2 is equal to, I don't know, I'm just going to put some random number a, and v3 is equal to b, then we can say, and then v1 would be equal to 1 half a plus 1 half b. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
If you divide by 4, you get v1 minus 1 half v2 minus 1 half v3 is equal to 0. Or v1 is equal to 1 half v2 plus 1 half v3. I just added these guys to both sides of the equation. Or we could say, if we say that v2 is equal to, I don't know, I'm just going to put some random number a, and v3 is equal to b, then we can say, and then v1 would be equal to 1 half a plus 1 half b. We can say that the eigenspace for lambda is equal to 3. Is the set of all vectors, v1, v2, v3, that are equal to a times v2 is a, right? So v2 is equal to a times 1. v3 has no a in it, so it's a times 0. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Or we could say, if we say that v2 is equal to, I don't know, I'm just going to put some random number a, and v3 is equal to b, then we can say, and then v1 would be equal to 1 half a plus 1 half b. We can say that the eigenspace for lambda is equal to 3. Is the set of all vectors, v1, v2, v3, that are equal to a times v2 is a, right? So v2 is equal to a times 1. v3 has no a in it, so it's a times 0. I'll save v1. Plus b times, v2 is just a, right? v2 has no b in it, so it's 0. v3 is 1 times, so 0 times a plus 1 times b. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So v2 is equal to a times 1. v3 has no a in it, so it's a times 0. I'll save v1. Plus b times, v2 is just a, right? v2 has no b in it, so it's 0. v3 is 1 times, so 0 times a plus 1 times b. And then v1 is 1 half a plus 1 half b. 1 half and 1 half. For any a and b, I could, you know, such that a and b are members of the reals. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
v2 has no b in it, so it's 0. v3 is 1 times, so 0 times a plus 1 times b. And then v1 is 1 half a plus 1 half b. 1 half and 1 half. For any a and b, I could, you know, such that a and b are members of the reals. Just to be a little bit formal about it. So that's our eigen, any vector that satisfies this is an eigenvector, and they're the eigenvectors that correspond to the eigenvalue lambda is equal to 3. So if you apply the matrix transformation to any of these vectors, you're just going to scale them up by 3. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
For any a and b, I could, you know, such that a and b are members of the reals. Just to be a little bit formal about it. So that's our eigen, any vector that satisfies this is an eigenvector, and they're the eigenvectors that correspond to the eigenvalue lambda is equal to 3. So if you apply the matrix transformation to any of these vectors, you're just going to scale them up by 3. Or you could say, let me write it this way. The eigenspace for lambda is equal to 3 is equal to the span, all of the potential linear combinations of this guy and that guy. So 1 half, 1 0, and 1 half, 0, 1. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So if you apply the matrix transformation to any of these vectors, you're just going to scale them up by 3. Or you could say, let me write it this way. The eigenspace for lambda is equal to 3 is equal to the span, all of the potential linear combinations of this guy and that guy. So 1 half, 1 0, and 1 half, 0, 1. So that's only one of the eigenspaces. That's the one that corresponds to lambda is equal to 3. Let's do the one that corresponds to lambda is equal to minus 3. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So 1 half, 1 0, and 1 half, 0, 1. So that's only one of the eigenspaces. That's the one that corresponds to lambda is equal to 3. Let's do the one that corresponds to lambda is equal to minus 3. So if lambda is equal to minus 3, I'll do it up here. I think I'll have enough space. Lambda is equal to minus 3. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Let's do the one that corresponds to lambda is equal to minus 3. So if lambda is equal to minus 3, I'll do it up here. I think I'll have enough space. Lambda is equal to minus 3. This matrix becomes, I'll do the diagonals, minus 3 plus 1 is minus 2. Minus 3 minus 2 is minus 5. Minus 3 minus 2 is minus 5. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Lambda is equal to minus 3. This matrix becomes, I'll do the diagonals, minus 3 plus 1 is minus 2. Minus 3 minus 2 is minus 5. Minus 3 minus 2 is minus 5. And then all the other things don't change. Minus 2, minus 2, 1, minus 2, minus 2, and 1. And then that times vectors and the eigenspace that corresponds to lambda is equal to minus 3 is going to be equal to 0. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Minus 3 minus 2 is minus 5. And then all the other things don't change. Minus 2, minus 2, 1, minus 2, minus 2, and 1. And then that times vectors and the eigenspace that corresponds to lambda is equal to minus 3 is going to be equal to 0. I'm just applying this equation right here, which we just derived from that one over there. So we're looking at the eigenspace that corresponds to lambda is equal to minus 3 is the null space of this matrix right here, or all the vectors that satisfy this equation. So the null space of this is the same thing as the null space of this in reduced row echelon form. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And then that times vectors and the eigenspace that corresponds to lambda is equal to minus 3 is going to be equal to 0. I'm just applying this equation right here, which we just derived from that one over there. So we're looking at the eigenspace that corresponds to lambda is equal to minus 3 is the null space of this matrix right here, or all the vectors that satisfy this equation. So the null space of this is the same thing as the null space of this in reduced row echelon form. So let's put it in reduced row echelon form. So the first thing I want to do, I'm going to keep my first row the same. I'm going to write a little bit smaller than I normally do, just because I think I'm going to run out of space. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So the null space of this is the same thing as the null space of this in reduced row echelon form. So let's put it in reduced row echelon form. So the first thing I want to do, I'm going to keep my first row the same. I'm going to write a little bit smaller than I normally do, just because I think I'm going to run out of space. So minus 2, minus 2, minus 2. And then, actually, let me just do it this way. I will skip some steps. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
I'm going to write a little bit smaller than I normally do, just because I think I'm going to run out of space. So minus 2, minus 2, minus 2. And then, actually, let me just do it this way. I will skip some steps. Let's just divide the first row by minus 2. So we get 1, 1, 1. And then let's replace this second row with the second row plus this version of the first row. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
I will skip some steps. Let's just divide the first row by minus 2. So we get 1, 1, 1. And then let's replace this second row with the second row plus this version of the first row. So this guy plus that guy is 0. Minus 5 plus minus, or let me say it this way. Let me replace it with the first row minus the second row, so minus 2 minus minus 2 is 0. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And then let's replace this second row with the second row plus this version of the first row. So this guy plus that guy is 0. Minus 5 plus minus, or let me say it this way. Let me replace it with the first row minus the second row, so minus 2 minus minus 2 is 0. Minus 2 minus minus 5 is plus 3. And then minus 2 minus 1 is minus 3. And let me do the last row. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Let me replace it with the first row minus the second row, so minus 2 minus minus 2 is 0. Minus 2 minus minus 5 is plus 3. And then minus 2 minus 1 is minus 3. And let me do the last row. I'll just do it in a different color for fun. And I'll do the same thing. I'll do this row minus this row. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And let me do the last row. I'll just do it in a different color for fun. And I'll do the same thing. I'll do this row minus this row. So minus 2 minus minus 2 is 0. Minus 2 plus 2. Minus 2 minus 1 is minus 3. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
I'll do this row minus this row. So minus 2 minus minus 2 is 0. Minus 2 plus 2. Minus 2 minus 1 is minus 3. And then we have minus 2 minus minus 5. So that's minus 2 plus 5. So that is 3. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Minus 2 minus 1 is minus 3. And then we have minus 2 minus minus 5. So that's minus 2 plus 5. So that is 3. Now let me replace, and I'll do it in two steps. So this is 1, 1, 1. I'll just keep it like that. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So that is 3. Now let me replace, and I'll do it in two steps. So this is 1, 1, 1. I'll just keep it like that. And actually, well, yeah, let me just keep it like that. And then let me replace my third row with my third row plus my second row. It'll just zero out. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
I'll just keep it like that. And actually, well, yeah, let me just keep it like that. And then let me replace my third row with my third row plus my second row. It'll just zero out. You just add these terms. These all just become 0. That guy got zeroed out. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
It'll just zero out. You just add these terms. These all just become 0. That guy got zeroed out. And let me take my second row and divide it by 3. So this becomes 0, 1, minus 1. And then I'm almost there. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
That guy got zeroed out. And let me take my second row and divide it by 3. So this becomes 0, 1, minus 1. And then I'm almost there. I'll do it in orange. So let me replace my first row with my first row minus my second row. So this becomes 1, 0. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And then I'm almost there. I'll do it in orange. So let me replace my first row with my first row minus my second row. So this becomes 1, 0. And then 1 minus minus 1 is 2. 1 minus minus 1 is 2. And then the second row is 0, 1, minus 1. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So this becomes 1, 0. And then 1 minus minus 1 is 2. 1 minus minus 1 is 2. And then the second row is 0, 1, minus 1. And then the last row is 0, 0, 0. So any v that satisfies this equation will also satisfy this guy. If this guy's null space, it's going to be the null space of that guy in reduced row echelon form. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And then the second row is 0, 1, minus 1. And then the last row is 0, 0, 0. So any v that satisfies this equation will also satisfy this guy. If this guy's null space, it's going to be the null space of that guy in reduced row echelon form. So v1, v2, v3 is equal to 0, 0, 0. Let me move this because I've officially run out of space. So let me move this lower down where I have some free real estate. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
If this guy's null space, it's going to be the null space of that guy in reduced row echelon form. So v1, v2, v3 is equal to 0, 0, 0. Let me move this because I've officially run out of space. So let me move this lower down where I have some free real estate. Let me move it down here. This corresponds to lambda is equal to minus 3. This was lambda is equal to minus 3, just to make us, you know, it's not related to this stuff right here. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So let me move this lower down where I have some free real estate. Let me move it down here. This corresponds to lambda is equal to minus 3. This was lambda is equal to minus 3, just to make us, you know, it's not related to this stuff right here. So what are all of the v1's, v2's, and v3's that satisfy this? So if we say that v3 is equal to t. If v3 is equal to t, then what do we have here? This tells us that v2 minus v3 is equal to 0. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
This was lambda is equal to minus 3, just to make us, you know, it's not related to this stuff right here. So what are all of the v1's, v2's, and v3's that satisfy this? So if we say that v3 is equal to t. If v3 is equal to t, then what do we have here? This tells us that v2 minus v3 is equal to 0. So it tells us that v2 minus v3, right? v2, 0 times v1 plus v2 minus v3 is equal to 0. Or that v2 is equal to v3, which is equal to t. That's what that second equation tells us. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
This tells us that v2 minus v3 is equal to 0. So it tells us that v2 minus v3, right? v2, 0 times v1 plus v2 minus v3 is equal to 0. Or that v2 is equal to v3, which is equal to t. That's what that second equation tells us. And then the third equation tells us, or the top equation tells us v1 times 1. So v1 plus 0 times v2 plus 2 times v3 is equal to 0. Or v1 is equal to minus 2v3, which is equal to minus 2 times t. So the eigenspace that corresponds to lambda is equal to minus 3 is equal to the set of all the vectors v1, v2, and v3, where, well, it's equal to t times v3 is just t, right? | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Or that v2 is equal to v3, which is equal to t. That's what that second equation tells us. And then the third equation tells us, or the top equation tells us v1 times 1. So v1 plus 0 times v2 plus 2 times v3 is equal to 0. Or v1 is equal to minus 2v3, which is equal to minus 2 times t. So the eigenspace that corresponds to lambda is equal to minus 3 is equal to the set of all the vectors v1, v2, and v3, where, well, it's equal to t times v3 is just t, right? v3 was just t. v2 also just ends up being t. So 1 times t. And v1 is minus 2 times t. For t is any real number. Or another way to say it is that the eigenspace for lambda is equal to minus 3 is equal to the span of the vector minus 2, 1, and 1. Just like that. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Or v1 is equal to minus 2v3, which is equal to minus 2 times t. So the eigenspace that corresponds to lambda is equal to minus 3 is equal to the set of all the vectors v1, v2, and v3, where, well, it's equal to t times v3 is just t, right? v3 was just t. v2 also just ends up being t. So 1 times t. And v1 is minus 2 times t. For t is any real number. Or another way to say it is that the eigenspace for lambda is equal to minus 3 is equal to the span of the vector minus 2, 1, and 1. Just like that. It looks interesting, because if you take this guy and dot it with either of these guys, I think you get 0. Is that definitely the case? If you take minus 2 times 1 half, you get a minus 1 there. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Just like that. It looks interesting, because if you take this guy and dot it with either of these guys, I think you get 0. Is that definitely the case? If you take minus 2 times 1 half, you get a minus 1 there. And then you have a plus 1. That's 0. And then minus 2 times 1 half. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
If you take minus 2 times 1 half, you get a minus 1 there. And then you have a plus 1. That's 0. And then minus 2 times 1 half. Yeah. You dot it with either of these guys, you get 0. So this line is orthogonal to that plane. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And then minus 2 times 1 half. Yeah. You dot it with either of these guys, you get 0. So this line is orthogonal to that plane. Very interesting. So let's just graph it, just so we have a good visualization of what we're doing. So we had that 3 by 3 matrix A. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So this line is orthogonal to that plane. Very interesting. So let's just graph it, just so we have a good visualization of what we're doing. So we had that 3 by 3 matrix A. It represents some transformation in R3. And it has two eigenvalues. And each of those have a corresponding eigenspace. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
So we had that 3 by 3 matrix A. It represents some transformation in R3. And it has two eigenvalues. And each of those have a corresponding eigenspace. So the eigenspace that corresponds to the eigenvalue 3 is a plane in R3. So this is the eigenspace for lambda is equal to 3. And it's the span of these two vectors right there. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And each of those have a corresponding eigenspace. So the eigenspace that corresponds to the eigenvalue 3 is a plane in R3. So this is the eigenspace for lambda is equal to 3. And it's the span of these two vectors right there. So if I draw, maybe they're like that, just like that. And then the eigenspace for lambda is equal to minus 3 is a line. It's a line that's perpendicular to this plane. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
And it's the span of these two vectors right there. So if I draw, maybe they're like that, just like that. And then the eigenspace for lambda is equal to minus 3 is a line. It's a line that's perpendicular to this plane. It's a line like that. It's the span of this guy. Maybe if I draw that vector, that vector might look something like this. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
It's a line that's perpendicular to this plane. It's a line like that. It's the span of this guy. Maybe if I draw that vector, that vector might look something like this. It's the span of that guy. So what this tells us, this right here is the eigenspace for lambda is equal to minus 3. So what that tells us, just to make sure we are interpreting our eigenvalues and eigenspaces correctly, is look, you give me any eigenvector in this. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |
Maybe if I draw that vector, that vector might look something like this. It's the span of that guy. So what this tells us, this right here is the eigenspace for lambda is equal to minus 3. So what that tells us, just to make sure we are interpreting our eigenvalues and eigenspaces correctly, is look, you give me any eigenvector in this. You give me any vector right here. Let's say that is vector x. If I apply the transformation, if I multiply it by a, I'm going to have 3 times that, because it's in the eigenspace for lambda is equal to 3. | Eigenvectors and eigenspaces for a 3x3 matrix Linear Algebra Khan Academy.mp3 |