Applications of 2x2 matrices | Wild Linear Algebra A 6 | NJ Wildberger

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 solving a system of linear equations; a family of problems
pg 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 coordinates 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)

Video Chapters:
00:00 Introduction
05:25 Arithmetic with 1×1 matrices / 2×2 matrices
11:54 Vector interpretation of a linear system
12:55 2×2 matrices
17:39 change of coordinates
21:17 Example. (AB)C=A(BC)
24:09 matrix notation; a column vector; matrix/vector multiplication
34:19 Four important matrices
44:21 determinants

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[alansvideoplex]

Really enjoy your videos. I watch them for fun! Best way to spend my stay at home during the Corona virus days. No one explains the why of linear algebra as well as you do.

[sacul]

Video contents

0:00 Intro
0:05 Arithmetic with 2x2 Matrices
2:11 Matrix Multiplication
5:24 Comparing Arithmetic in 1x1 and 2x2 Matrices
10:58 The Determinant Function for 1x1 Matrices
12:54 The Determinant Function for 2x2 Matrices
17:21 Exercise: Proving the Laws of Matrix Arithmetic
21:16 Example: Proof of the Associative Law for 2x2 Matrices
26:32 Geometry of Linear Transformations
28:53 Examples of Linear Transformations 1
31:37 Examples of Linear Transformations 2 (Composition of Transformations)
34:19 Four Important Matrices
37:23 Examples of Linear Transformations 3
38:37 Examples of Linear Transformations 4
40:23 Definition of a Linear Transformation