{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}