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)
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