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

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

View Full Document Right Arrow Icon
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,
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 02/18/2010 for the course MATH 51 taught by Professor Staff during the Fall '07 term at Stanford.

Ask a homework question - tutors are online