midterm_1_exam - L in the form y = mx . b. Two of these...

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MA 35100 GOINS MIDTERM EXAM #1 This exam is to be done in 50 minutes in one continuous sitting. Collaboration is not allowed, but you may use your own personal notes; the lecture notes posted on the course web site; homework solutions posted on the course web site; and Bretscher’s Linear Algebra with Applications . Write your answers on a separate sheet of paper. Provide all details of your work : either give precise references for or proofs of any statements you claim. Problem 1. Consider the following linear system of equations: ± ± ± ± x + 2 y = 5 3 x + 5 y = 13 ± ± ± ± a. Find the augmented matrix A corresponding to the system. b. Compute its reduced row-echelon form E = rref( A ). c. Find the solution of the linear system. Problem 2. Consider the following 2 × 2 matrices: A = ² 0 - 1 1 0 ³ , B = ² 0 1 1 0 ³ , and C = ² 0 1 - 1 0 ³ . a. One of these matrices represents a reflection about a line L through the origin. Ex- plain which it is, and give an explicit equation for
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Unformatted text preview: L in the form y = mx . b. Two of these matrices represent rotations in the coordinate plane. Explain which they are, and give their corresponding angles of rotation. c. Compute A 2 , B 2 , and C 2 . Give geometric interpretations of your answers. Problem 3. a. Consider the intersection of two lines: a 11 x + a 12 y = b 1 a 21 x + a 22 y = b 2 Give an example of two lines which do not intersect, i.e., an example of a linear system in two variables which is inconsistent. b. The projectivization of a system is dened as the homogeneous system a 11 x 1 + a 12 x 2-b 1 x 3 = 0 a 21 x 1 + a 22 x 2-b 2 x 3 = 0 Prove that two projective lines always intersect. In fact, show that the projectiviza-tion of a system has innitely many solutions. 1...
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This note was uploaded on 03/31/2012 for the course MA 351 taught by Professor ?? during the Spring '08 term at Purdue University-West Lafayette.

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