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Unformatted text preview: AMS 210: Applied Linear Algebra October 29, 2009 AMS 210 Topics Today Problem Set 8 Midterm 2 Linear Transformations and AffineLinear Transformations A Theorem on Linear Transformation Theorems on Affine Linear Transformations Examples of Composition From 2D to 3D and back Model Transforms View Transforms Projection Transforms Processing a Point through the Pipeline AMS 210 Problem Set 8 This week’s problem set will not be collected or graded due to the upcoming exam. You will, however, be responsible for its content. Read sections 3.4 and 3.5. Exercises: 3.4: 8b, 11iab, 15, 16; 3.5: 2, 6a, 7, 18, 19ad, 22. AMS 210 Midterm 2 November 5 during class. Covers all of chapter 3: sections 3.1, 3.2, 3.3, 3.4, and 3.5 (including but not exhausively, determinants, Gaussian elimination, GaussJordan pivoting, inverses and their uses, iterative solutions, and condition numbers). November 3 class will be a review lecture with no new material. Scientific calculators, plus TI83/84/86, allowed. TI89class calculators (TI89/92, HP48/49/50/etc) disallowed. Example problems on Blackboard, as are problem set solutions through problem set 6. AMS 210 Linear Transformations and AffineLinear Transformations Linear transformations map points in a space to other points in a space. For computer graphics purposes, the spaces of interest tend to be either 2D or 3D Euclidean 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. From 2D to 3D and back 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. T 3 : x = x + y and y = y : applies 45 ◦ clockwise skew to y axis AMS 210 A Theorem on Linear Transformations Let T be a linear transformation with w = T ( w ) and v = T ( v )....
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 Fall '09
 ShuaiXue
 Linear Algebra, aﬃne linear transformations

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