Linear Algebra 4f: Linear Subspaces of Polynomials

[xoppa]

A linear algebra approach to differential equations makes studying differential equations a lot more interesting (and easier to remember). Thanks teacher.

[dishankdazzler]

All of lecture series in this channel are really amazing. I have been wanting to learn graduate level maths more intuitively as I did for my school high school maths. But there are not many resource out there like that. I would say these videos are really insightful.

[darrenpeck]

Incredible that linear invariants givr rise to subspaces and are so simply captured with primary school algebra =0 even though complicated mathematics is being used. What an absolute triumph of simplicity and generality. Feels like a Fundamental language

[danielmoss]

Thank you so much! This video has been far more useful than any of my lectures from my professors at university!

[rovshanabdurrahimov]

This Professor is king of math. wow i m following all the videos one by one andgiving like. THANK YOU VERY VERY MUCH

[rd-tkjs]

very insightfull covering a lot in just 10 minutes. thanks !

[tangolasher]

Hi, thanks for all your help. You explain the definition of linear properties very clearly for geometric vectors and Rn, however, you really gloss over the meaning of linear properties in this 3rd category, polynomials. You gloss over basic polynomials and immediately jump to differential equations - a huge jump that left me completely dumbfounded! Not everyone is so comfortable with calculus! Could you please make a video explaining the meaning of linear properties of polynomials more thoroughly, with much more simple polynomials first (quadratic, or even more basic ones)? Until you do that, I am unable to continue with your course, due to a lack of understanding of this 3rd piece of the puzzle. It's a shame, because it was so enlightening up until this video! Thanks.

[mrnarason]

Thanks, I was given a hw problem like this I had no idea how to answer it

[stephanewamba]

thank you very much for these excellent videos, i am wondering though what textbook you recommend.

[stgr]

Hi,
These videos are great! Thank you so much! You are a great teacher and you seem sincerely interested in trying to help and I really appreciate that. With that in mind, I have the following (hopefully constructive) suggestions for this video:
From a pedagogical standpoint, I suspect this video needs to be broken into 2 parts (each of which is about as long as the current video): the first part could be functions in general and the second part could be polynomials.
Also, even though YOU see closure very clearly without working out the integral in the 2nd example or the derivatives in the 3rd example, there are likely many Linear Algebra students who have had calculus who would see closure more readily if you actually worked out a VERY simple example of each case. Notice that you, yourself, briefly tried to find a solution to the differential equation at 6:50 (e^1/3x) as you were explaining things. I found myself doing the same thing back at the integral problem. When you explain polynomials, you end up working out the integral and the derivatives (very quickly) anyway. Had you previously worked through this, the polynomial explanation would benefit greatly.
Finally, the third example (the differential equation) doesn't seem to benefit much from having the scalar value (3) on the first term unless you are going to explain in more detail that the scalar does not alter closure. Without more explanation, I think the scalar distracts a bit from the point you're making.

Thanks again!
Steve

[richardaversa]

So your example polynomial must have coefficients a=0, b=0? Interesting... This leads to the only polynomial that is a solution to that differential equation being the constant function C. I suppose indeed this is true (though it is a trivial solution for that differential equation)

[antonellomascarello]

1:39 IMPORTANT: Test for Lin Property
2:03 Example #1 -> 9:42 with polynomials
2:59 Example #2 -> 10:45 with polynomials
6:18 Example #3 -> 11:40 with polynomials
8:18 9:21 Focus on polynomials
8:35 IMPORTANT : No Common template for Functions (but possible with polynomials)

[xoppa]

5:00 Can we include more general types of functions such that f(1)=0 such as continuous functions, or general real valued functions?

[michaellewis]

First could be written more directly as {f(x)|f(1)=0}