Linear Algebra 12a: Applications Series - Polynomial Interpolation

[@MathTheBeautiful]

Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.

[@Steve-3P0]

OK. This is was VERY good about WHY we want to use interpolation and linear algebra. THANK YOU! I kept asking my Linear Algebra teacher 20 years ago "Why do I want to learn this?" His answer was always "We learn math for spiritual growth".

[@knivesoutcatchdamouse2137]

To be fair, your professor was partially right!

[@shofibahalwan402]

Okay.. This is very good. Thank you!! Subbed and liked...

[@КлинтИствуд-ъ3и]

Amazing series!!

[@oliverhulesch3922]

always good for offering insight

[@deantodd6845]

You just blessed me.

[@mybankofamerica]

BTW I your videos are really good. I really appreciate these.

[@dishankdazzler]

How is ax+bx^2+cx^3+dx^4 is linear function? Isn't it a polynomial? I get that you say it's linear in a,b,c,d. But that's what not linear refers to in general? being Linear means in the changing variable which x here. a,b,c,d are just coefficients here.

[@MathTheBeautiful]

You're right, it's not a linear function. But that's the beauty of it that many nonlinear things have linear elements in them.

[@loremazza5191]

Given n points, is there always a polynomial that passes through all of them?

[@denisgiard130]

very good video ; but just a little detail : the portrait shown cannot possibly be Vandermonde's. He lived in the second half of the 18th century ; and the dress and haircut of the man on the portrait are obviously around 1600.

[@mybankofamerica]

Hi, I was wondering what software you use when you are inserting the matrix and picture

[@MathTheBeautiful]

Bill Johnson Scientific Workplace

[@vineetmukim2365]

We can say A is invertible AFTER looking at the columns, how can you say it beforehand? Am I missing something?

[@MathTheBeautiful]

Which point in the video are you referring to?

[@vineetmukim2365]

time 06:33

[@MathTheBeautiful]

Yes, not known at that point. I should have said "will have proven to be invertible".

[@vineetmukim2365]

Thanks a lot for replying, Sir.

[@eduardoschiavon5652]

I guess it follows from the theorem that there exists a unique polynomial of degree less than or equal to n that interpolates a given function in n+1 data points.

[@D3r3k2323]

I don't understand why this is linear algebra, since the polynomial is not linear.

[@MathTheBeautiful]

Hi Derek, That's actually the whole point! Polynomials are not linear but we are combining them in linear combinations. Sines and cosines are not linear other, but Fourier series are pure linear algebra! I hope this is helpful.

[@debendragurung3033]

You are correct on the fact that the equation of polynomials of variate y on variate x is not linear. But thats bit of a pre college definition of calling line as linear, polynomial of degree quadratic 2 and so on.. But the equation of straight y =ax+b is just some equation thats also functions. The axioms of linear Mapping are F(au) = aF(u) F(u+v) = F(u) + F(v) if you consider u and v as variate x, even the linear function y=ax+b doesnt obey these axioms of Linear Mapping. Well thats where I once got stucked. But if viewed from other perspective of function space, if two functions of any degree continous on open interval of x, f(x) +g(x) = (f+g)(x) (af(x)) = a(f(x) for constant a f(x) and g(x) can be any functions polynomials or nonpolynomials like y=e^x , y= 1/x etc . As long as they are functions they obey axioms of linear Mapping. This is the foundation for Linear Algebra. I dont know how Linear descended, but I believe its from how we solve system of linear equations like intersection of two lines, then intersection of three planes etc. It basically came from linear equations, to linear functions then to Linear Mapping.