WildLinAlg6: Applications of 2x2 matrices

This is the 6th lecture in this course on Linear Algebra. We review matrix arithmetic for 2x2 matrices and compare the main laws with the simpler case of 1x1 matrices. Then we apply 2x2 matrices to study transformations of vectors, with examples including reflections, rotations and dilations.

We visualize these with pictures using the standard Cartesian coordinate framework. The concept of a linear transformation is also introduced.

This is part of a series on Linear Algebra given by N J Wildberger, also the discoverer of Rational Trigonometry.

• Category

  Education

[frobeniusfg]

Two vectors define a lattice in 2D (lattices are extensively used in crystallography). Vectors can be packed into a matrix. An inverse matrix will be composed then from the transposed vectors of the primitive vectors of the reciprocal lattice. That could lead to a gentle introduction to the co- and contra- variant coords.

[njwildberger]

Quite right, this is a nice geometric interpretation of the inverse which I will hopefully explain at some point.

[imaginalrealm]

very nice, thanks.

[EmptySpaceEnterprise]

pg 1: @00:08

2x2 matrix; indicies notation vs direct notation; operations: scalar multiplication, addition;

pg 2: @02:13

matrix multiplication; 1x1 matrix; 2x2 matrix as a "generalized number";

pg 3: @05:25

rules governing arithmetic with matrices;

pg 4: @10:59

determinant of matrix; Inverse law; proof;

pg 5: @12:55

identity matrix; inverse law for 2x2 case; notation for the inverse matrix;

[EmptySpaceEnterprise]

pg 6: @17:22 exercises 6.(1:2) ; example: The distributive law for scalar multiplication; pg 7: @19:44 example proof continued; pg 8: @21:17 another example proof; The associative law for matrix multiplication; pg 9: @23:34 example proof continued; geometry of plane transformations @26:00; pg 10: @26:32 linear transformations here we come; introducing the standard coordinate system; pg 11: @28:54 example 1: reflection in x-axis; example 2: reflection in line y=x ;

[EmptySpaceEnterprise]

pg 12: @31:37

example 3: rotation by a half-turn; composition of transformations;

pg 13: @34:21

naming 4 important matrices; formation of a closed group under multiplication; remark about link of matrix multiplication and complex numbers @36:38 ;

pg 14: @37:25

example: dilation by a factor 2; example: non-uniform dilation;

pg 15: @38:39

example: shear;

pg 16: @40:25

Fix a matrix A; define a 'mapping'; Main Fact:a linear transformation; suggested exercises @43:13 ;