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EN_lzzqr9xGaeQ_SPEAKER_00
Whatever this is, this is some matrix, right? If this matrix is invertible, so in other words, if the inverse matrix exists,
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EN_lzzqr9xGaeQ_SPEAKER_00
Right. So the second video is going to show you a method for finding the eigenvalues and eigenvectors of a general square matrix. So the way it works is first of all, I tell you how you find the eigenvalues of that matrix. And then next you find the eigenvectors. So it goes in two steps. So first of all, I explain how...
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EN_lzzqr9xGaeQ_SPEAKER_00
I can write this as A times V is lambda times I times V, where I is the identity matrix, which does nothing, right? Then I can take this over the other side and then I get A minus lambda I times V should be equal to zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
The non-zero V is if this condition fails, right? The only way that we can find a non-zero V satisfying this equation is if the inverse does not exist. If the inverse exists, then the solution must be zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
The fact that A minus lambda I minus one does not exist.
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EN_lzzqr9xGaeQ_SPEAKER_00
Because one of our conditions on eigenvector, eigenvectors is that they should not be zero, right? We want a non zero V, which satisfies this equation. So the only way we can get a non zero C.
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EN_lzzqr9xGaeQ_SPEAKER_00
But this means that V is not an eigenvector.
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EN_lzzqr9xGaeQ_SPEAKER_00
Then I can multiply both sides by the inverse matrix, and we get that A minus lambda I minus one, A minus lambda I B equals A minus lambda I minus one, zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
But when does the inverse not exist? We showed last week that you can always find the inverse as long as the determinant is not zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
So the defining equation for eigenvectors and eigenvalues was that the matrix A times the eigenvector V should be equal to lambda.
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EN_lzzqr9xGaeQ_SPEAKER_00
So the only time the investors not exist is if the determinant is zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
This gives you a polynomial equation in lambda. So this is.
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EN_lzzqr9xGaeQ_SPEAKER_00
Okay. So in order to have an eigenvector, this equation must be true. The determinant of A minus lambda times I must be equal to zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
Okay, so we can use this equation to find the eigenvalues.
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EN_lzzqr9xGaeQ_SPEAKER_00
Okay. So we have to solve a polynomial equation, in other words. Okay. And this equation is called the characteristic polynomial.
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EN_lzzqr9xGaeQ_SPEAKER_00
is equal to A, that's this, minus lambda times I is this.
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EN_lzzqr9xGaeQ_SPEAKER_00
So that's how you find the eigenvalues. So I'm going to do an example of this because I think this may have been a bit abstract. So we'll work on the same example we saw in the first video.
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EN_lzzqr9xGaeQ_SPEAKER_00
The determinant of A minus lambda I is the determinant of this. That's 4 minus lambda.
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EN_lzzqr9xGaeQ_SPEAKER_00
minus, minus 12 plus 6 minus 6. Okay. And in order for the eigenvalue eigenvector to exist, this must be equal to zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
was that A minus lambda I times the eigenvector V should be equal to zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
And here, from the previous sheet, A minus lambda I was the matrix.
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EN_lzzqr9xGaeQ_SPEAKER_00
n in lambda. Okay, so it's like lambda to the power n plus something times lambda to the n minus one plus all the way down to zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
minus 3 minus lambda minus minus 1 times 6, which is lambda squared minus lambda
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EN_lzzqr9xGaeQ_SPEAKER_00
And what's useful about it is if you solve this equation, then you know that lambda must be an eigenvalue.
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EN_lzzqr9xGaeQ_SPEAKER_00
Which is four minus lambda minus one, six minus three minus lambda. Okay, so that's a two by two matrix. And according to this equation, we must have determined that matrix is zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
And in this case, it factorizes quite nicely. This is lambda. Minus.
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EN_lzzqr9xGaeQ_SPEAKER_00
So if we say that V is equal to XY, then this means that one minus one, six minus six times XY is equal to zero zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
Okay, so the solution you need here is obviously the x equals y.
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EN_lzzqr9xGaeQ_SPEAKER_00
So we need to solve this equation for each of the possible eigenvalues that we found. So the first one is that lambda 1 was 3. So if you put lambda equals 3 into this equation,
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EN_lzzqr9xGaeQ_SPEAKER_00
Which means, so you get two equations, x minus y is zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
Lambda 2, which is minus 2. Okay. And again, this agrees with what we saw in the previous video.
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EN_lzzqr9xGaeQ_SPEAKER_00
That's the characteristic polynomial, and you see a two by two matrix, you get a degree two polynomial, which is a quadratic equation. Okay?
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EN_lzzqr9xGaeQ_SPEAKER_00
It works on any size of matrix. The only difference is in for larger matrices, you have a more difficult determinant to calculate more equations to solve here.
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EN_lzzqr9xGaeQ_SPEAKER_00
So this shows you the general method for calculating eigenvectors and eigenvalues.
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EN_lzzqr9xGaeQ_SPEAKER_00
So, in the next video, I'm just going to give some more examples of this for different kinds of matrix.
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EN_lzzqr9xGaeQ_SPEAKER_00
Well, first of all, for an n by n matrix,
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EN_lzzqr9xGaeQ_SPEAKER_00
Okay, so that means that v1 equals 1, 1 is the eigenvector.
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EN_lzzqr9xGaeQ_SPEAKER_00
And then you get the V2. X is 1, Y is 6X.
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EN_lzzqr9xGaeQ_SPEAKER_00
Okay. And this is true because this is exactly what we saw in the first video, right? If you multiply this vector by A, you get three times this vector. So that's good. It shows you the method works.
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EN_lzzqr9xGaeQ_SPEAKER_00
Y is six X. And again, I'm free to choose what X or Y is. So again, I'll take X equals one.
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EN_lzzqr9xGaeQ_SPEAKER_00
So again, I'll do it a bit quicker this time. A minus lambda I times V equals zero. Let's call this one V2.
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EN_lzzqr9xGaeQ_SPEAKER_00
So quickly, we'll just do it for the second eigenvalue. That's lambda two, which was minus two.
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EN_lzzqr9xGaeQ_SPEAKER_00
X to be whatever you want. So I'm going to choose X equals one. So that means that the back to be one.
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EN_lzzqr9xGaeQ_SPEAKER_00
Again, the two equations are the same. This means that 6x minus y is zero.
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EN_lzzqr9xGaeQ_SPEAKER_00
6x minus 6y is zero. And you'll notice, hopefully, that these two equations are the same. That's not a coincidence. If it's a real eigenvalue, then these two equations will all always be the same, up to a multiple of a constant.
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EN_lzzqr9xGaeQ_SPEAKER_00
And the equation that we derived for the eigenvectors
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EN_lzzqr9xGaeQ_SPEAKER_00
Three times lambda plus two equals zero. So there are only two solutions. Lambda one is three or lambda two is minus two. So the only possible eigenvalues this matrix can have are three or minus two.
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EN_l-Ui117TohA_SPEAKER_00
You'll find your turns become smoother, more consistent, as you go along here.
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EN_l-Ui117TohA_SPEAKER_00
And if you like, you can pick goals to aim for, like the stripes out on the runway here.
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EN_l-Ui117TohA_SPEAKER_00
Now let's work into a little bit of forward flight. Bring your helicopter up into a hover.
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EN_lBxlC71G5U0_SPEAKER_01
just had some apple slices, peanut butter, my dick.
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EN_l-Ui117TohA_SPEAKER_00
Put a little bit of left in, and you can start flying back the other way.
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EN_l-Ui117TohA_SPEAKER_00
Bring it to a hover over here. Go ahead and turn the nose of the helicopter to the left a little bit.
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EN_lBxlC71G5U0_SPEAKER_01
I'm all over the place, can you tell? Ah, class starts in a minute. This is from Nina and Alexa, by the way. Thank you. Oh gosh.
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EN_l-Ui117TohA_SPEAKER_00
As you get more and more comfortable with this, you'll find yourself moving out farther and farther, and faster and faster. That's alright, just take your time doing it.
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EN_xjiBM1WIHbM_SPEAKER_02
Yeah, right. I mean, it's summertime. There's got to be some other getaway. No. Yeah. They're going, they're going. This could be one of my favorite clips I'm about to play, by the way, of all time. They're going to Washington, DC. And why?
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EN_lBxlC71G5U0_SPEAKER_01
like, wake up again, but it's really nice because then when I have my laptop propped up on this book stand, then it doesn't make me like strain my eyes or back or ruin my posture to reach for the mouse or keyboard. I think I am literally crazy for not getting it sooner. Yeah, I'll definitely recommend getting like a B...
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EN_lBlr0rCAmwU_SPEAKER_02
And then, get on muddy. Korean style omelets. Cause I love eggs and Joan's mommy knows how much I love eggs.
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EN_xjiBM1WIHbM_SPEAKER_02
And luckily, the state constitution of Texas has
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EN_xjiBM1WIHbM_SPEAKER_02
She's filibustering Republicans in Texas. She got on a plane.
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EN_xjiBM1WIHbM_SPEAKER_02
You think I could only call one special session? I will call special session after special session after special session, which is very difficult to say, by the way. Uh, and he said, we're just going to keep going. We're going to keep going until they come back and they do the job that they were hired to do. And they ...
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EN_xjiBM1WIHbM_SPEAKER_01
Using a filibuster to flee the state of Texas to plead with the president to do away with the filibuster in Washington D.C. That is the example of hypocrisy on its face.
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EN_xjiBM1WIHbM_SPEAKER_02
There are going to be Texas state police officers or Texas, Texas Rangers. I don't even know who does the arresting. There was just some kind of posse will be there. Greg Abbott sitting on a horse with a big old hat and a shotgun. And they're gonna be like, all right.
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EN_xjiBM1WIHbM_SPEAKER_02
She's going to Washington DC to talk to a guy who has no power over filibustering to get rid of the filibuster while she herself is in the middle of a filibuster that I'm pretty sure she has no idea is even happening. This is, I mean, this is going on around the country, but nowhere has it become more, more ridiculous...
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EN_xjiBM1WIHbM_SPEAKER_02
They have a failsafe plan, folks. They were messing around. When they wrote the state of Texas Constitution, they thought maybe something like this might happen. So they have a contingent.
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EN_xjiBM1WIHbM_SPEAKER_02
Are the Texas Democrats who are filibustering and fleeing the state to deny quorum? Why are they going to Washington D.C.? Representative, I'm sorry, Democrat Representative Jasmine Crockett posted a video explaining exactly why. Listen to this.
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EN_lCuYZI37SZc_SPEAKER_01
And then the other ones are like covered in this red powder. These are pretty bland. The stuff with the red powder is really, really shrimpy. It's good.
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EN_xjiBM1WIHbM_SPEAKER_01
Isn't that the most un-Texan thing you've ever heard? Texans running from a fight? They're, they're quitters. It's like during a football game or a baseball game, taking their equipment when they're way behind and just leaving the field.
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EN_lBxlC71G5U0_SPEAKER_01
Pursue therapy and external therapists would probably be best instead of from the institution that is.
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EN_xjiBM1WIHbM_SPEAKER_02
So when they don't get their way, they don't negotiate. They don't try to come to some kind of agreement. They get on a bus, they grab some beer because they're still Democrats. They love the party. And then they get on a Cush private jet that was probably paid for by either A, the taxpayers or B, some rich donor, lik...
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EN_xjiBM1WIHbM_SPEAKER_02
But you stay and fight in Texas. Uh, but, you know, I mean, I guess fight or flight. They're like, we'll take, we'll take the flight. It's a private one. Um, seems like it should be, uh, should be pretty good. But Laura Ingraham, he can, she continued to press Governor Abaddon. She's like, look, what are you going to ...
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EN_y3soebij4UI_SPEAKER_07
believe in you. Battle what? Fight. Hm.
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EN_lCuYZI37SZc_SPEAKER_00
It's not, it's not that spicy. It's just really shrimpy. As it says, it's a shrimp chip. So it's going to be very shrimpy and fishy. So it's six out of 10. And Sky gave it a five. So.
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EN_xjiBM1WIHbM_SPEAKER_02
Unfortunately, it didn't work out for them. But, you know, that's the same, that happens sometimes in life. It happens with, the Democrats should stay and fight. They're gonna lose.
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EN_lCuYZI37SZc_SPEAKER_01
Eh, it smells like fish food. It smells like what I feed my fish.
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EN_AGXAgKsIXDu_SPEAKER_01
The price is also very affordable. You can use my discount link and code down below and get an 83% off and three months for free.
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EN_IrThiTDr-fY_SPEAKER_03
How you make me feel. But wondering if it's the same for you. But we can't let around.
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EN_AGXAgKsIXDu_SPEAKER_01
And of course, your IP will not be exposed as well. Surfshark has over 3,200 servers in 95 different countries. Especially for Netflix users, using VPN would be super easy to switch from country to country. So for those of you who wants to learn another language, Surfshark VPN would be very useful.
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EN_xjiBM1WIHbM_SPEAKER_01
The State House of Representatives who are here in the Capitol in Austin right now, they do have the ability to issue a call to have their fellow members who are not showing up to be arrested, but only so long as that arrest is made in the state of Texas. That's why they have fled the state. Once they step back into t...
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EN_AGXAgKsIXDu_SPEAKER_01
Also, one account can unlimitedly be shared and downloaded by different devices. Isn't that amazing?
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EN_lVrTgiPD6CU_SPEAKER_01
could take on a different level of propulsion. If you have prepared yourself well
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EN_lVrTgiPD6CU_SPEAKER_00
I know there's like some special sadhana that is being given in the center for people who are below 33. So I wanted to know, like, if I pass this mark, what's the significance in my spiritual path?
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EN_lVrTgiPD6CU_SPEAKER_01
Either with education or some skill or yoga or inner dimensions of your life.
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EN_lCuYZI37SZc_SPEAKER_01
More fish stuff. I think I'm really getting, like, my palate is sort of adjusting to this stuff. Like, I used to not be crazy about showing up and eating a bunch of fish flavored stuff.
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EN_lVrTgiPD6CU_SPEAKER_01
Your physiological structures, your energy structures, your internal geometry, if you brought it to a certain level of maturity by the time you're thirty-three, then life takes off much more effortlessly. Otherwise, it will need much more effort.
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EN_lVrTgiPD6CU_SPEAKER_01
Your childhood in many ways ends at eight-and-a-half. So if you take two size solar cycles, which is twenty-four years or little less than that, and add eight-and-a-half years to it, it will become
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EN_lVrTgiPD6CU_SPEAKER_01
See, you want to throttle up now and fire up really big, but your tank is empty.
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EN_lVrTgiPD6CU_SPEAKER_01
This is why we set up sadhana for below thirty-three years of age, so that they prepare
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EN_lCuYZI37SZc_SPEAKER_01
That's really not, they have some that are like just regular, like pop looking trim things.
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EN_AGXAgKsIXDu_SPEAKER_01
Do you want to know how to avoid internet hackers and to protect your personal information? The answer is using VPN. Today I will introduce Surfsharp VPN. What is VPN? The main purpose of a VPN?
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EN_lVrTgiPD6CU_SPEAKER_01
46 may become more troublesome than possibilities, but there are many possibilities. When a woman reaches 46 years of age,
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EN_lVrTgiPD6CU_SPEAKER_01
You know, in the western societies, they talk about midlife crisis. It usually starts around thirty-three, thirty-four years of age.
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EN_lVrTgiPD6CU_SPEAKER_01
Not that you can never fly, that's a wrong thing to say, but difficult. You need more fuel to fly. You will need lot more fuel to fly, but if you've brought your life,
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EN_lVrTgiPD6CU_SPEAKER_01
A whole lot of people burst forth at this time, but by then lot of people are sick, they are, they got various problems, and they,
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EN_lVrTgiPD6CU_SPEAKER_01
But for all this, you must at least keep yourself in a reasonable shape and form. If you're trying to imitate Mother Earth, or you become sick,
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EN_lVrTgiPD6CU_SPEAKER_01
As I've spoken in recent times, this is only now, in the last sixty to seventy years, that you have to stick a certificate in your face and go, and that stupid certificate opens doors for you.
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EN_lVrTgiPD6CU_SPEAKER_01
Otherwise, never in the history of humanity, a certificate opened a door. It is only you, your competence, which opened the door. Once again, world is moving that way.
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EN_lVrTgiPD6CU_SPEAKER_01
Well, when that moment comes, possibilities open up naturally and many things are possible.
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EN_lVrTgiPD6CU_SPEAKER_01
If you have prepared yourself on any of these things, that is the time when it can propel or that is the time when it can start crashing.
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EN_lVrTgiPD6CU_SPEAKER_01
And fortunately, as you become adult somewhere, at least you have to keep one hand in the marketplace, because the whole world has become economics. But you threw eleven year old child, that means at eleven if he became CEO, probably he started at six.