Unformatted text preview: Solutions to quiz # 7 (October 21)
1. The cosine of the angle between a + b and b is
a+b Â·b
a+b Â· b . Now,
a + b Â· b = a Â· b + b Â· b and a+b Â· a+b = a+b = a Â· a + 2a Â· b + b Â· b. We have
aÂ·a= a 2 = 4, bÂ·b= b 2 =9 and a Â· b = a Â· b Â· cos angle between a and b = 2 Â· 3 Â·
Therefore,
a + b Â· b = 2 + 9 = 11, a+b = âˆš 1
= 2.
3 4+2Â·2+9=
âˆš
and the cosine of the angle between a + b and b is 11/3 17. âˆš 17 Answer: the cosine of the angle between a + b and b is
11
âˆš.
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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 Fall '08
 Conger
 Linear Algebra, Algebra, Vector Space, a+b, orthogonal projection, a+b Â· a+b

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