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Unformatted text preview: AMS 210: Applied Linear Algebra November 17, 2009 AMS 210 Topics Today Chapter 5. . . Linear Transformations Abstract Linear Transformations Eigenvectors as Natural Coordinate Systems for Transformations Eigenfunctions of Linear Transformations AMS 210 Chapter 5. . . . . . differs from previous chapters insofar as it’s not proximately exampledriven  it’s much more theoretical and akin to a traditional linear algebra course. AMS 210 Linear Transformations Already seen these in lecture on linear algebra in computer graphics. Since there was no homework on this, will review (modified) in lecture. AMS 210 Linear Transformations and AffineLinear Transformations Linear transformations map points in a space to other points in a space (but these don’t have to be easil visualisable spaces). All linear transformations are representable via matrices: where T ( w ) is a linear transformation of the point w , there exists a matrix A such that T ( w ) = Aw . Affine linear transformations are slightly more general and can be represented as w = Aw + b . Affine linear transformations also lack some useful properties than linear transformations. AMS 210 Examples of Transformations T 1 : x = 2 x + 4 and y = 3 y + 2: doubles width, triples height, and translates by (4, 2). T 2 : x = cos 45 ◦ x sin 45 y ≈ . 707 x . 707 y and y = sin 45 ◦ x + cos 45 ◦ y ≈ . 707 x + 0 . 707 y : rotates 45 degrees counterclockwise....
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 Fall '09
 ShuaiXue
 Eigenvectors

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