Assignment 4, MATH 240, Spring 2008.
 Due 9:30am, Thursday February 28th.
 Please post your solutions in the dropoff box outside the lab AQ 4135.
 Sorry, no late assignments accepted.
 As usual, I've included some odd numbered questions.
 Please hand in your solution to 3.1 #19, 3.2 #9b, and 27.  

 From the text: Fraleigh & Beauregard (3rd edition)
 2.3 exercises 8,21,22,29
 3.1 exercises 1,2,10,12,18,19
 3.2 exercises 1,2,9b,12,20,22,25,26,27,28
 3.3 exercises 4,7,8,10
 7.1 exercises 8,9,12a

 Additional exercises:

 1: Consider the linear transformation T:R^3 -> R^2 given by
    T(x,y,z) = (2 x + y + z, x + y + 3 z).
    Give its standard matrix representation.
    What points in R^3 are mapped by T to the origin in R^2.
    What is this set of points called?

 2: Consider the linear transformation T:R^2 -> R^2 given by
    T(x,y) = (rx,-y) for r>1.  Describe what T does to R^2 either in words
    or with a sketch.  Since T is a linear transformation we can write
    T(x) = A x where A is a 2 by 2 matrix and x is in R^2.  What is A?
    Is T invertible?  If not, explain why.  If so, give the inverse.

 3:            [cos t  -sin t]
    The matrix [             ] rotates the XY plane anticlockwise by t degrees.
               [sin t   cos t]
    Write down the standard matrix representations for
    T1(x) = A x that rotates the XY the plane by +45 degrees and
    T2(x) = B x that rotates the XY the plane by -45 degrees and
    verify that AB=I.  See page 150 of section 2.3.
 
 4: Let v1,v2,v3 be vectors in R^n.
    Let U = Span(v1,v2,v3) and W = Span(v1,v2,v3,v4=v1+2v2).
    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.
 
 5: Is the set of even functions from R to R is a vector space?
    If yes give a proof.  If no give a counter example.
    Note: f(x) is an even function if f(-x) = f(x) for all x in R.
    Some examples of even functions are 1, x^2, and cos x.
 
 6: Write out the definitions for
    (i) subspace of a vector space,
    (ii) linear independence of a set of vectors in a vector space, and
    (iii) basis for vector space.
    Don't hand in.

 Notes on exercises:
 For exercise 28 of 3.2 it will help if you first think geometrically.
 What geometric object is W1 in R^3?  What is W2 in R^3? 
 Hence what geometric object must the intersection of W1 and W2 be?

 For exercise 12 of 7.1 let v = [1,1,1].  Calculate also vB and vB'.
 Now verify that your change of basis matrix C from B to B' is correct
 by checking that C vB = vB'.