Professor MathTheBeautiful, thank you for using Linear Algebra techniques when solving Linear ODEs with Constant Coefficients. This is an error free video/lecture on RUclips.
I am a math graduate student just brushing up on ODEs and you are the first person who ever explicitly explained where the characteristic polynomial comes from. It's such a simple thing too... Making a note to self to remember to take the time to just show that one step to my students in the future.
it is such a smart way of explaining in using null space. I learned ODE the traditional way back at school, but it the linear algebra way is way more straightforward.
All the actual solving was very simple and matched well what I remember learning in DE courses, but I really liked how it was kept general and linked to linear operators on a vector space. And I agree that "separate and integrate" is a nice trick, but too limited, and even if a DE is separable, it isn't always obvious how.
Funny, in my linear algebra class we never studied the concept of 'null space', and I haven't readily seen it in the few books I have, and yet in set, group, and ring theory we did study the kernel of maps, which seems to be an equivalent kind of sub-structure. I am diagnosed on the ASD scale (Asperger's) and suffer a lot of internal stress when open questions are floored about who can 'see the solution', I like how you make it alright not to know just yet though. I think you promise often never to place people in direct spot-light with questions, that is excellent IMO.
Yes, kernel and null space are 100% synonymous and interchangeable. I do advise my students often to "see" the solution. The point is not "this is so easy, anyone can see the solution", but "this is the kind of problem where, with practice, you should learn to see the solution".
Prof grinfeld - thank you for your pointer. I went and listened to the "most important 30 minutes in my mathematical life" and at minute 19:55 realized why I hesitated. Somehow, beause of your example with a 3x4 matirx, I had it in the back of my mind that had the matrix been a 2x4 then the argument wouldn't have worked BUT obviously this is wrong. The number of rows determines the size of the null space but the fact that any solution can be obtained from one particular solution plus an appropriate vector in the null space remains true. Thank you... but I have another question in an upcaming lecture!
Two remarks: A) I think that λ should be factored to -2 and -4 (the sum of which is -6 and the product +8) and B) it seems to me that you explain very rapidly - maybe it was done previously - why another particular solution of the homogenized equation (sounds like milk - now the variables and the coefficinets don't separate...sorry couldn't resist) if it exists would not result in another "dimension" of the null space. I know that you are correct in your formulation of the general solution but I missed being convinced that there is no other particilar solution to the homogenized equation that is not already spanned by C1e-2t + C2e-4t
Hi George, I believe that it must in this video: ruclips.net/video/VfJN9Px3KyM/видео.html that I talk about that, although there are many particular solutions, *any* *one* will do! Pavel PS: I'm not 100% sure of the point you make in A)
09:23 Although I like factoring too, it only works with "school" problems where the numbers are pretty and whole, but not with "real-life" problems in which the solutions are often irrational :q And good luck with that with higher-degree polynomials :q 16:03 Well, at least for the constant-coefficient ones. Non-constant coefficients are whole another story... :q
Sci Twi, well, good for you, but that's not the point. It's just an example of a solution existing. It is up to you to find those solutions by whatever means necessary.
Professor MathTheBeautiful, thank you for using Linear Algebra techniques when solving Linear ODEs with Constant Coefficients. This is an error free video/lecture on RUclips.
I am a math graduate student just brushing up on ODEs and you are the first person who ever explicitly explained where the characteristic polynomial comes from. It's such a simple thing too... Making a note to self to remember to take the time to just show that one step to my students in the future.
it is such a smart way of explaining in using null space. I learned ODE the traditional way back at school, but it the linear algebra way is way more straightforward.
All the actual solving was very simple and matched well what I remember learning in DE courses, but I really liked how it was kept general and linked to linear operators on a vector space.
And I agree that "separate and integrate" is a nice trick, but too limited, and even if a DE is separable, it isn't always obvious how.
Awesome lecture , you do explain the method and also why we do it , a deeper understanding, also you are teaching us linear algebra at the same time
Funny, in my linear algebra class we never studied the concept of 'null space', and I haven't readily seen it in the few books I have, and yet in set, group, and ring theory we did study the kernel of maps, which seems to be an equivalent kind of sub-structure. I am diagnosed on the ASD scale (Asperger's) and suffer a lot of internal stress when open questions are floored about who can 'see the solution', I like how you make it alright not to know just yet though. I think you promise often never to place people in direct spot-light with questions, that is excellent IMO.
Yes, kernel and null space are 100% synonymous and interchangeable.
I do advise my students often to "see" the solution. The point is not "this is so easy, anyone can see the solution", but "this is the kind of problem where, with practice, you should learn to see the solution".
Prof grinfeld - thank you for your pointer. I went and listened to the "most important 30 minutes in my mathematical life" and at minute 19:55 realized why I hesitated. Somehow, beause of your example with a 3x4 matirx, I had it in the back of my mind that had the matrix been a 2x4 then the argument wouldn't have worked BUT obviously this is wrong. The number of rows determines the size of the null space but the fact that any solution can be obtained from one particular solution plus an appropriate vector in the null space remains true. Thank you... but I have another question in an upcaming lecture!
How can a school have such a great teacher and such terrible chalk?
Two remarks: A) I think that λ should be factored to -2 and -4 (the sum of which is -6 and the product +8) and B) it seems to me that you explain very rapidly - maybe it was done previously - why another particular solution of the homogenized equation (sounds like milk - now the variables and the coefficinets don't separate...sorry couldn't resist) if it exists would not result in another "dimension" of the null space. I know that you are correct in your formulation of the general solution but I missed being convinced that there is no other particilar solution to the homogenized equation that is not already spanned by C1e-2t + C2e-4t
Hi George, I believe that it must in this video: ruclips.net/video/VfJN9Px3KyM/видео.html that I talk about that, although there are many particular solutions, *any* *one* will do!
Pavel
PS: I'm not 100% sure of the point you make in A)
I used the Laplace Transform to solve both of the equations on 7:30 and I got the exactly same answers. xD
6:46 I said “Half Life” I’m sorry
That was not half bad
09:23 Although I like factoring too, it only works with "school" problems where the numbers are pretty and whole, but not with "real-life" problems in which the solutions are often irrational :q And good luck with that with higher-degree polynomials :q
16:03 Well, at least for the constant-coefficient ones. Non-constant coefficients are whole another story... :q
Sci Twi, well, good for you, but that's not the point.
It's just an example of a solution existing. It is up to you to find those solutions by whatever means necessary.