WildLinAlg5: Change of coordinates and determinants
This is the 5th lecture of this course on Linear Algebra. We analyse the fundamental problem of inverting a change of coordinates, and give applications to solving a system of two linear equations in two unknowns. This problem has two interpretations, one in terms of meets of lines, and one in terms of combining vectors to form a given vector.We introduce vectors and matrices in a purely algebraic way. Composing two changes of variable leads to the basic multiplication formula for matrices. This has a close connection with determinants.This course is given by Assoc Prof N J Wildberger of UNSW, also the discoverer of Rational Trigonometry, which is explained in the WildTrig series.CONTENT SUMMARY: pg 1: @00:08 inverting the relationship between two pairs of variables;pg 2: @02:59 application of change of coordinates;pg 3: @05:43 changing coordinates is related to soloving a system of linear equations; a family of problemspg 4:? @10:06 example with parallel lines; zero determinant then 2 lines parallel (2-dim);pg 5: @11:54 Vector interpretation of a linear system;pg 6: @15:45 example of 2 vector system without solution;pg 7: @17:39 change of coordinants as heart of the subject;pg 8: @21:26 generalize the example on the previous page;?pg 9: @24:09 matrix notation; a column vector; matrix/vector multiplication;pg 10: @26:20 writing a pair of linear equations in matrix/vector form;pg 11: @29:34 arithmetic with matrices and vectors; introducing notation independent of application; you might think of this page as the start of a course in linear algebra; column vectors, scalar multiplication,? addition, subtraction; approach is independent of any geometric interpretation;pg 12: @32:26 laws of vector arithmetic;pg 13: @33:42 geometrical interpretation of abstract column vectors;pg 14: @35:19 A matrix as an array of numbers; scalar multiplication, addition, subtraction; matrices follow the same laws as vectors;pg 15: @38:37 define product of matrix and a column vector; define product of two 2by2 matrices;pg 16: @41:45 examples;pg 17: @44:21 determinants; alternate notation; theorem: determinant of product of matrices is equal to the product of the determinants of 2 matrices; exercises 5.(1:2)? ;pg 18: @46:19 exercises 5.(3:5) ; (THANKS to EmptySpace Enterprise)
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