# Linear Algebra

The University of New South Wales,

Updated On 02 Feb, 19

The University of New South Wales,

Updated On 02 Feb, 19

This course on Linear Algebra is meant for first year undergraduates or college students. It presents the subject in a visual geometric way, with special orientation to applications and understanding of key concepts. The subject naturally sits inside affine algebraic geometry. Flexibility in choosing coordinate frameworks is essential for understanding the subject. Determinants also play an important role, and these are introduced in the context of multi-vectors in the sense of Grassmann. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry.

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4.1 ( 11 )

This lecture studies spaces of polynomials from a linear algebra point of view. We are especially interested in useful bases of a four dimensional space like P^3 polynomials of degree three or less. We introduce the standard (or power) basis, also the modified Factorial basis. Translations of the corresponding functions yield linear transformations, giving Taylor bases and a purely algebraic definition of the derivative. We see that some basic calculus ideas are really algebraic in nature, not requiring `real numbers, limits or slopes of tangents.

This course in Linear Algebra is given by N J Wildberger.

CONTENT SUMMARY pg 1 @0008 Map of a space of polynomials to a space of vectors; Definition linearvector space; course distinction @0327 ;

pg 2 @0446 Definition An ordered basis of a linear space; Example1 ;

pg 3 @0703 Example2 (basis of a vector space); Example3 (basis of a polynomial space); Definition Dimension of a linearvector space; examples;

pg 4 @0925 The space of polynomials is richer than the isomorphic space of vectors; translating polynomials @0950 ; degree of the polynomial is preserved in translation;

pg 5 @1320 Study01 of translation by 3 (see previous page); A linear transformation @1615 ;

pg 6 @1648 study01 continued; image and kernel of polynomial of degree 3 @1716;

pg 7 @1904 the derivative of a function appears in translation;

pg 8 @2249 Definition The derivative of a polynomial; calculus via linear algebra @2300;

pg8_Theorem; factorial notation;

pg 9 @2805 Calculus as algebra; pg9_Theorem (product rule); proof (Leibniz mentioned);

pg 10 @3346 Translating a polynomial and obtaining the derivatives; Taylor series mentioned;

pg 11 @3758 Importance of various bases; standard basis; factorial basis; Example ; coefficient vectors of a polynomial with respect to a basis;

pg 12 @4210 Theorem (Basis isomorphism correspondence); The standard vector space of column vectors @4530 ;

pg 13 @4622 The derivative as a linear transformation; remark about formulas in calculus and combinatorics @5046 ;

pg 14 @5113 Another basis; the standard basis moved over by 3; example;

pg 15 @5432 Change of basis matrix; How to get this matrix! @5456 ;

pg 16 @5716 Exercises 20.1-3; closing remarks @5827; (THANKS to EmptySpaceEnterprise)

Sam

Sep 12, 2018

Excellent course helped me understand topic that i couldn't while attendinfg my college.

Dembe

March 29, 2019

Great course. Thank you very much.