Differential Equations and exp (At)

[@quirkyquester]

Great explanation, thank you so much!

[@rohitn6333]

very very good lecture mam and thanks a lot for this one . I was kinda confused with the concepts taught in the previous couple of lectures but this recitation completely cleared all my confusion

[@quantfund2002]

Great teaching thank you

[@uranium-h3o]

me: in my way to solve systems of differential equations. degree 3 polynomial: no ...

[@chuzai2156]

still confused why in f(t), x1, x2,x3 only chose those vectors' last element.

[@mauriciobarda]

at 1:34 when she creates the u-vector y is the third element of it.

[@chuzai2156]

@@mauriciobarda thanks，bro

[@azaz868azaz5]

does x2 must be 1,-1,-1?

[@wasp4932]

good job

[@dibeos]

What if I get a matrix S that has no inverse?

[@yifangu5604]

By the definition, S is made up of a linearly independent set of vectors. By a linearly independent set of vectors, we mean a set of vectors {v_1, v_2, ..., v_n} meets the following requirement: c_1*v_1+c_2*v_2+...+c_n*v_n=0, where c_1=c_2=...=c_n=0. Since S=[v_1|v_2|...|v_n], we can rewrite the combination " c_1*v_1+c_2*v_2+...+c_n*v_n=0" into the matrix multiplication, which is Sc=0 (c is a column vector). It is impossible to type a column vector here. I will present c in such a way: the column vector c= the transpose of the row vector [c_1, c_2, ..., c_n]. Notice that the only way to make Sc=0, by the definition of a linearly independent set of vectors, is that all components of c are 0's, which means vector c is a zero vector. All in all, it turns out that the zero vector c is a unique solution of Sc=0. Recall that if a system Ax=b has a unique solution, then A has an inverse. Hence, S must have an inverse. Q.E.D

[@yifangu5604]

I did not make it clear that S is made up of a linearly independent set of eigenvectors.

[@yifangu5604]

As long as eigenvalues are distinct, their corresponding eigenvectors are linearly independent.

[@nabeelkhan7506]

How did we go from u(t) to y(t)

[@alagappank1242]

1:32 y at 3rd coordinate

[@mayuridas5161]

I might be wrong but just as per the lectures I do not think so the eigenvalues turned out to be correct ones.

[@fanzhang3746]

I don't understand what is `exp(At)`, and why only the first column is interesting? I thought `At` is just a mathematical trick to get u(t) ...

[@fanzhang3746]

Oh, I got it, because the first column corresponds to the expression of y''', which IS the problem.

[@zewenliu9280]

​@@fanzhang3746 Take it easy mate, there is no such disguised purpose. Frankly speaking, calculating the first column is just a practice for determinant and inverse matrix, which are we really need to concerned. Why it looks similar to that y equation? ------> u(t) can be written as C*exp(At).

[@JthElement]

What?? What a random, comment.@@zewenliu9280

[@hswu2288]

ah 你还会微分方程，在国内念那个学校

[@piupolino2618]

This is pretty common worldwide, might be google translate mixing it up but I assume she didn't learn it in China