This course on linear algebra with polynomials aims to teach the application of linear algebra concepts to spaces of polynomials. The learning outcomes include understanding polynomials, polynomial functions, interpolation, regression in statistics, and the connection between polynomials and linear algebra. The course covers topics such as finding polynomials through points, regression techniques, and the use of isomorphism to relate polynomial spaces to vector spaces. The teaching method involves a geometric approach to linear transformations. This course is intended for learners interested in mathematics, particularly those looking to deepen their understanding of linear algebra and its applications in polynomial spaces and regression analysis.
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
