More applications of 2x2 matrices | Wild Linear Algebra A 7 | NJ Wildberger

This is the 7th lecture in this course on Linear Algebra. Here we continue discussing 2x2 matrices, their interpretation as linear transformations of the plane, how to analyse rotations, including a rational formulation, and how to combine rotations and reflections.

Finally we discuss the connections with calculus, introducing the idea that the derivative is really a linear transformation.

Video Contents:
00:00 A bit of review; matrix/vector multiplication; define a mapping/function/transformatio­n; A linear transformation
02:40 Proof of transformation linearity
05:25 Rule implied by knowledge of linearity; mapped base vectors; area dilation factor
10:37 Determining how the basis vectors transform; The columns of the transformation matrix are the transformations of the basis vectors
11:56 Examples
14:44 Example continued; rotations; unit circle; rotation matrix
19:14 Rotations by 30degree's, 45degree's, 60degree's
23:49 Some trig identities; exercise 3.1
26:28 Rational parametrization; alternate rotation matrix; exercise
30:22 Reflection
35:11Rreflection continued; Composition of linear transformations
38:02 Example: rotation/reflection composition
39:24 Example continued
42:44 Linear approximations to non-linear maps; globally nonlinear/locally approx-linear; differential calculus mentioned
45:50 Example of linear approx. to non-linear map; Leibniz's notation
50:28 Example continued; the derivative matrix at a point; Lesson derivatives are linear transformations @51:19
52:18 Exercises 7.3-4
53:27 Exercises 7.5-7 ; (THANKS to EmptySpaceEnterprise)

************************

Here are the Insights into Mathematics Playlists:

Here are the Wild Egg Maths Playlists (some available only to Members!)

[huntsman]

I could never remember the damn Trignometry formulas (cos(60) = 1/2 etc). All that changed now. The vector form of the Trignometry formulas shown at 21.12 are simply imprinted in my understanding. Thank you so much, Professor Wilberger!

[alpozen]

oh man sir you are doing a service to humanity with these series!

[ys]

OMG these videos actually make you understand the true essence of linear algebra and actually have a profound understanding of what these things mean rather than just memorizing ways to solve for bases, determinants, crammer's rule etc

[jamesfinch]

These lectures have helped a lot with my understanding of rigid body dynamics. Thank you.