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math-51-f09-hw4-extras

math-51-f09-hw4-extras - assume that they cancel each other...

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MATH 51, FALL 2009 PROBLEM SET 4 — EXTRA PROBLEMS E1. Let P be a plane in R 3 that is spanned by two perpendicular vectors v and w . Assuming that the lengths of those vectors are equal to one, do the following: a) express the result of orthogonal projection onto the plane P as a linear com- bination of vectors v and w . b) find the matrix of the orthogonal projection onto the plane P in terms of coordinates of vectors v and w . E2. Suppose two islands are inhabited by birds. Each year, 1/4 of the birds on island 1 move to island 2, and the other 3/4 remain on island 1. Similarly, each year 1/3 of the birds on island 2 move to island 1, and the other 2/3 remain on island 2. (For this problem, you should ignore births and deaths, or, if you prefer,
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Unformatted text preview: assume that they cancel each other out.) Let v ( n ) be the vector in R 2 whose i-th entry is the number of birds on island i in the year n . 1. Find a matrix A so that v ( n + 1) = A v ( n ) 2. Find all vectors v in R 2 such that A v = v . What is the signiﬁcance of such a vector v in terms of bird relocations? E3. There are 6000 undergraduate students at Stanford. Let M be the 6000 × 6000 matrix whose ij entry is 1 if student i and student j have met each other and 0 if they have not met. What information does M 2 contain? That is, what is the meaning of the ij entry of M 2 ? 1...
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