Explore the applications of 2x2 matrices in this comprehensive 55-minute lecture from the Wild Linear Algebra series. Delve into the interpretation of 2x2 matrices as linear transformations of the plane, analyze rotations including rational formulations, and examine the combination of rotations and reflections. Gain insights into the connections between linear algebra and calculus, introducing the concept of derivatives as linear transformations. Learn through detailed examples, proofs, and exercises covering topics such as matrix-vector multiplication, linearity properties, basis vector transformations, rotation matrices, trigonometric identities, rational parametrization, reflections, and composition of linear transformations. Conclude with an introduction to linear approximations of non-linear maps and the derivative matrix, bridging the gap between linear algebra and differential calculus.
Overview
Syllabus
CONTENT SUMMARY: pg 1: @ a bit of review; matrix/vector multiplication; define a mapping/function/transformatioÂn; A linear transformation;
pg 2: @ proof of transformation linearity;
pg 3: @05:25 rule implied by knowledge of linearity; mapped base vectors; area dilation factor @;
pg 4: @ determining how the basis vectors transform; The columns of the transformation matrix are the transformations of the basis vectors;
pg 5: @ examples;
pg 6: @ example continued; rotations; unit circle; rotation matrix;
pg 7: @ rotations by 30degree's, 45degree's, 60degree's
pg 8: @ some trig identities; exercise 3.1
pg 9: @ Rational parametrization; alternate rotation matrix; exercise
pg 10: @ reflection
pg 11: @ reflection continued; Composition of linear transformations
pg 12: @ example: rotation/reflection composition
pg 13: @ example continued;
pg 14: @ Linear approximations to non-linear maps; globally nonlinear/locally approx._linear; differential calculus mentioned;
pg 15: @ example of linear approx. to non-linear map; Leibniz's notation;
pg 16: @50:28 example continued; the derivative matrix at a point; Lesson derivatives are linear transformations @ ;
pg 17: @ exercises 7.3-4 ;
pg 18: @ exercises 7.5-7 ; THANKS to EmptySpaceEnterprise
Taught by
Insights into Mathematics
