Spring 2007 - Mengi's Class - Quiz 2

Spring 2007 - Mengi's Class - Quiz 2 - Quiz 2 Math 20F...

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Quiz 2, Math 20F - Lecture B (Spring 2007) 1. A linear transformation T : R 2 R 2 maps the four points and the line segments joining them displayed on the left to the ones displayed on the right in the figure below. 4 0.5 1 1.5 2 2.5 3 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 a) (2.5 points) Find the 2 × 2 matrix A associated with this linear transformation, that is find an A such that T ±² x y ³´ = A ² x y ³ . Specifically T ±² 1 3 ³´ = ² 2 . 5 3 ³ , T ±² 1 2 ³´ = ² 2 2 ³ , T ±² 1 1 ³´ = ² 1 . 5 1 ³ , and T ±² 2 1 ³´ = ² 2 . 5 1 ³ . Solution: This is just a shear transformation. You should notice that the vertical com- ponents remain the same, but horizontally the L-shape moves to the right. The higher the point is located, the more it moves to the right. Therefore the transformation should be in the form T ±² x y ³´ = ² x + cy y ³ = ² 1 c 0 1 ³² x y ³ where c is a positive constant. It is clear from the change in the horizontal components that c = 0 . 5. For instance the horizontal components of the points
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This note was uploaded on 04/23/2008 for the course MATH 20F taught by Professor Buss during the Spring '03 term at UCSD.

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Spring 2007 - Mengi's Class - Quiz 2 - Quiz 2 Math 20F...

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