Differential Equations and exp (At)
HTML-код
- Опубликовано: 5 фев 2025
- MIT 18.06SC Linear Algebra, Fall 2011
View the complete course: ocw.mit.edu/18...
Instructor: Linan Chen
A teaching assistant works through a problem on differential equations.
Watch this video in Chinese: • 微分方程指数矩阵 (At)
License: Creative Commons BY-NC-SA
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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
Great explanation, thank you so much!
Great teaching thank you
me: in my way to solve systems of differential equations.
degree 3 polynomial: no ...
does x2 must be 1,-1,-1?
What if I get a matrix S that has no inverse?
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
I did not make it clear that S is made up of a linearly independent set of eigenvectors.
As long as eigenvalues are distinct, their corresponding eigenvectors are linearly independent.
good job
still confused why in f(t), x1, x2,x3 only chose those vectors' last element.
at 1:34 when she creates the u-vector y is the third element of it.
@@mauriciobarda thanks，bro
How did we go from u(t) to y(t)
1:32 y at 3rd coordinate
I might be wrong but just as per the lectures I do not think so the eigenvalues turned out to be correct ones.
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) ...
Oh, I got it, because the first column corresponds to the expression of y''', which IS the problem.
@@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).
What?? What a random, comment.@@zewenliu9280
ah 你还会微分方程，在国内念那个学校
This is pretty common worldwide, might be google translate mixing it up but I assume she didn't learn it in China