Linear algebra with polynomials | Wild Linear Algebra A 19 | NJ Wildberger

Spaces of polynomials provide important applications of linear algebra. Here we introduce polynomials and the associated polynomial functions (we prefer to keep these separate in our minds).

Polynomials are vital in interpolation, and we show how this works. Then we explain how regression in statistics (both linear and non-linear) can be viewed using our geometric approach to a linear transformation.

Finally we discuss the use of `isomorphism' to relate the space of polynomials up to a certain fixed degree to our more familiar space of column vectors of a certain size.

CONTENT SUMMARY: pg 1: @00:08 Linear algebra applied to polynomials; polynomials;
pg 2: @03:33 a general polynomial; associated polynomial function; example;
pg 3: @07:35 importance of polynomial functions;
pg 4: @10:37 Interpolation;
pg 5: @12:23 finding a polynomial going through one point/two points; example; pg 6: @14:44 example continued;
pg 7: @18:11 example (find the line through 2 points);
pg 8: @20:47 (find the polynomial through 3 points); Vandermonde matrix @22:40 ; the pattern @24:11;
pg 9: @25:02 Regression (statistics); looking for an approximate solution;
pg 10: @26:59 Regression continued;
pg 11: @30:09 Linear regression; remark on the power of linear algebra @32:39;
pg 12: @33:04 Spaces; the connection between polynomials and linear algebra; operations; similarity of polynomials and vectors;
pg 13: @35:48 trying to say this object is like this object; mapping: start out with a polynomial and end up with a vector of coefficients @37:24 ; isomorphism; vector of coefficients; bijection @38:07 ; surjective; injective;
pg 14: @40:46 connection between functions and an abstract 3d vector space;
pg 15: @43:36 Exercises19.1-3;
pg 16: @44:51 Exercise 19.4; (THANKS to EmptySpaceEnterprise)

Video Chapters:
00:00 Introduction
3:33 A polynomial determines a polynomial function
7:35 Importance of polynomial functions
10:36 Interpolation
12:22 One point
14:44 Augmented matrix approach
20:45 Three points polynomial through
25:02 Regression
30:07 Linear regression
33:03 Polynomial spaces
40:46 Connection between functions and an abstract 3d vector space
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Here are the Insights into Mathematics Playlists:

[njwildberger]

Yes we will be discussing this in the second half of the course, which has more applications, in particular geometrical ones.

[KlikPatrick]

3AM and I finally found some good material to understand polynomial space. Wonderful. By the way, nice drumstick.

[njwildberger]

This is more complicated, since multiplication of polynomials raises degree, so we can't just consider a space like P^3. But there is of course a study of this multiplication: the polynomials form what is called a commutative algebra.

[vishwanathteggihalli]

Its really wonderful, the way you see the complex concepts, and help millions of people fall in love with Mathematics...the best part of your lecture series is that you know exactly what and when these concepts have to be taught....Tysm Sir!!

[mustafamohammed]

very good, very impressive, thank you very much

[travisdriessen]

I thought finding the determinant of the 3x3  Vandermonde was tricky. I had to come to terms with column operations being a "legal move". In the end what convinced me was the notion that the determinant of the transpose = the determinant (so columns can be treated essentially as rows). After this the next useful tool was factoring rows out of the matrix (note for others: columns can be factored also). After these steps I went back to row reduction to get into upper triangular form and then the determinant fell out. 

Now I just need to come to terms with what this means for finding the unique second degree polynomial for a given set of points! 

[AbstruseGoose]

This lecture was great, the explanation of regression was very pleasing. Please tell me we get an explanation of finding the closest projection to the image in a later lecture.