solq7 - Solutions to quiz # 7 (October 21) 1. The cosine of...

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Unformatted text preview: Solutions to quiz # 7 (October 21) 1. The cosine of the angle between a + b and b is a+b ·b a+b · b . Now, a + b · b = a · b + b · b and a+b · a+b = a+b = a · a + 2a · b + b · b. We have a·a= a 2 = 4, b·b= b 2 =9 and a · b = a · b · cos angle between a and b = 2 · 3 · Therefore, a + b · b = 2 + 9 = 11, a+b = √ 1 = 2. 3 4+2·2+9= √ and the cosine of the angle between a + b and b is 11/3 17. √ 17 Answer: the cosine of the angle between a + b and b is 11 √. 3 17 2. The image of the orthogonal projection of the plane onto line a is the line a itself, so rank A = 1. Similarly, the image of the orthogonal projection of the plane onto line b is the line b itself, so rank B = 1. We have (AB )x = A (Bx) for all vectors x from R2 . If x is not perpendicular to b then its orthogonal projection Bx onto b is a non-zero vector and, since lines a and b are not perpendicular, the orthogonal projection of Bx onto line a is also a non-zero vector. This proves that the image of the linear transformation with matrix AB is the line a and hence rank AB = 1. b x a 0 Answer: rank A = 1, rank B = 1 and rank AB = 1. 1 ...
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This note was uploaded on 12/26/2011 for the course MATH 214 taught by Professor Conger during the Fall '08 term at University of Michigan.

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