00:02
We'll look at question number 16.
00:07
And we're going to consider the two vectors x of t and y of t here.
00:13
And the question is whether or not they are linearly independent or linearly dependent.
00:21
Okay.
00:22
So now there's numerous ways to check this or there's different ways to solve it.
00:29
But there's either going directly from the definition or using the ronskin.
00:37
And in this video, i'm going to use the ronskin.
00:43
So, okay, why use the ronskin? i can use the ronskin because the matrix x of t is square.
00:55
Okay.
00:57
And the idea of what the ronskin is, is if there's a value of t in the interval i, such that w of t is not zero, the vectors are linearly independent.
01:15
And i should have mentioned that the ronskin, the ronskin here, w of t, w of t, w of t, w, of t is equal to the determinant of x of t so the determinant of x of t so the determinant of x of t where x of t is the matrix created by the um the vectors you wish to check that are linearly independent or not um so i can use the round skin here because x of t is square.
02:11
So in other words, i have two, i have two, two by two vectors that i wish to check linear independence.
02:21
If you look at another problem in the textbook, you'll, you have three two by two vectors which in that case you can't use the round scan, because the resultant matrix x of t would not be square.
02:43
And i'll be doing that question next.
02:46
That's question number 18.
02:47
So that's a case where you can't use the round scheme.
02:51
But in this case, you can because x of t is square.
02:57
Okay.
02:57
So now in this case, we have x of t is.
03:09
This matrix, it's the matrix spanned by the, it's the matrix generated by the two vectors, x of t.
03:17
This is x of t and this is y.
03:25
And the ravsky and then of x is equal to the determinant of x of t.
03:33
And that is equal to then just using directly the calculation of the determinants.
03:40
We get we get 16 here so okay so now this is this is my determinant remember the way you calculate the determining of a two by two matrix you just use you you multiply this well this is this is sine t times cosine 2t minus these two so it's basically this times this minus this times this and now that's a that's a trick you should have learned earlier in the linear algebra stuff again if you if you don't remember or don't know this linear algebra material you should check the earlier sections or find a book on linear algebra to get that material.
04:43
Okay, next, a couple formulas that you should know from calculus.
04:53
A very important formulas to know, which is that sine 2t, these two formulas both sine 2t and cosine 2t, particularly the second one cosine 2t it can get dressed different ways but either way any way you dress it up it's still basically this formula and really this is the one i want to use because it simplifies the algebra in the end and so now i'm just going to apply the algebra using both the formulas i told you to know.
05:37
And so this is directly the calculation of the ronscan.
05:41
And now this step here, i'm just replacing, i'm replacing cosine 2t with the formula from before.
05:53
So i'm replacing that here.
05:55
And this is this second part here.
05:59
I'm just replacing, again, i'm just using the replacement for sign 2t, and here i'm just replacing the cosine 2t there.
06:12
And next, this next step, what i'm doing is i'm factoring out.
06:20
So what you'll notice in the previous formula is there's a sign t here...