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WildLinAlg7: More applications of 2x2 matrices

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.CONTENT SUMMARY: pg 1: @00:08 a bit of review; matrix/vector multiplication; define a mapping/function/transformatio?n; A? linear transformation;pg 2: @02:40 proof of transformation linearity;pg 3: @05:25 rule implied by knowledge? of linearity; mapped base vectors; area dilation factor @09:20;pg 4: @10:37 determining how the basis vectors transform; The columns of the transformation matrix are the transformations of the basis vectors;pg 5: @11:56 examples;pg 6: @14:44 example continued; rotations; unit circle; rotation matrix;pg 7:? @19:14 rotations by 30degree's, 45degree's, 60degree'spg 8: @23:49 some trig identities; exercise 3.1pg 9: @26:28 Rational parametrization; alternate rotation matrix; exercisepg 10: @30:22 reflectionpg 11: @35:11 reflection continued; Composition of linear transformationspg 12: @38:02 example: rotation/reflection compositionpg 13: @39:24 example continued;pg 14: @42:44 Linear approximations to non-linear maps; globally nonlinear/locally approx._linear; differential calculus? mentioned;pg 15: @45:50 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 @51:19 ;pg 17: @52:18 exercises 7.3-4 ;pg 18: @53:27 exercises 7.5-7 ; (THANKS to EmptySpaceEnterprise)
Length: 55:12


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