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WildLinAlg19: Linear algebra with polynomials

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)
Length: 46:14


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