Assignment 3, MATH 240, Spring 2008.
 Due 5:30pm, Friday February 15th.
 Please post your solutions in the dropoff box outside the lab AQ 4135.
 Sorry, no late assignments will be accepted.

 From the text Fraleigh & Beauregard

 1.7 exercises 22, 28, 32, 40.
 2.1 exercises 1, 4, 10, 22, 26, 28, 30, 32
 2.2 exercises 6, 11.
 2.3 exercises 2, 3, 4, 5, 13, 14

 Note, I've assigned 6 odd numbered exercises.  Most of their
 answers are in the back of the book.  You don't have to hand them in.


 Additional exercises:

 1: Calculate the steady-state distribution vector s
    (the vector s such that T s = s whose entries sum to 1)
    for the transition matrix T shown in 1.7 exercise 34.

 2: Let W = { [x,y,z] : x + y = 0 and x,y,z are in R }.
    Since x + y = 0 is the equation of a plane through
    the origin in R^3, W is a subspace of R^3.
    Sketch W in R^3, find a basis for W, and include your
    basis vectors in the sketch.

 3: Let v1,v2,v3 be vectors in R^n.
    Let U = Span(v1,v2,v3) and W = Span(v1,v2,v3,v4=v1+v2).
    Prove that U = W by showing
     (i) every vector in U is also in W and
    (ii) every vector in W is also in U.