Explore the application of linear algebra to polynomials in this 46-minute lecture. Delve into the world of polynomials and their associated functions, understanding their vital role in interpolation. Learn how to find polynomials passing through specific points and discover the power of the Vandermonde matrix. Examine regression in statistics from a geometric perspective, focusing on both linear and non-linear approaches. Investigate the concept of isomorphism, connecting polynomial spaces to familiar column vector spaces. Gain insights into the relationship between functions and abstract vector spaces. Practice with provided exercises to reinforce your understanding of these fundamental concepts in linear algebra and polynomial theory.
Overview
Syllabus
CONTENT SUMMARY: pg 1: @ Linear algebra applied to polynomials; polynomials;
pg 2: @ a general polynomial; associated polynomial function; example;
pg 3: @ importance of polynomial functions;
pg 4: @ Interpolation;
pg 5: @12:23 finding a polynomial going through one point/two points; example; pg 6: @ example continued;
pg 7: @ example find the line through 2 points;
pg 8: @20:47 find the polynomial through 3 points; Vandermonde matrix @22:40 ; the pattern @;
pg 9: @ Regression statistics; looking for an approximate solution;
pg 10: @ Regression continued;
pg 11: @30:09 Linear regression; remark on the power of linear algebra @;
pg 12: @ 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 @ ; surjective; injective;
pg 14: @ connection between functions and an abstract 3d vector space;
pg 15: @ Exercises19.1-3;
pg 16: @ Exercise 19.4; THANKS to EmptySpaceEnterprise
Introduction
A polynomial determines a polynomial function
Importance of polynomial functions
Interpolation
One point
Augmented matrix approach
Three points polynomial through
Regression
Linear regression
Polynomial spaces
Connection between functions and an abstract 3d vector space
Taught by
Insights into Mathematics
