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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 nonzero vector
and, since lines a and b are not perpendicular, the orthogonal projection of Bx onto line
a is also a nonzero 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.
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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.
 Fall '08
 Conger
 Linear Algebra, Algebra

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