Assignment 5, MATH 240, Spring 2008.
 Due 9:30am, Tuesday March 11th.
 Please post your solutions in the drop-off box outside the lab AQ 4135.
 Please hand in solutions for the odd numbered exercises which don't
 have solutions in the back of the textbook.
 Sorry, the "no late assignments accepted" policy stands.

 From the text Fraleigh & Beauregard
 7.1 exercise 16
 3.4 exercises 4,8,35.
 4.2 exercises 4,6,18,20,21,26,30,31,32
 4.3 exercises 8,26,35(a)-(f)

 For exercise 4 of 4.2, use the method of minor expansion from section
 4.2 to calculate the determinant of the given matrix and also also use
 the Gaussian elimination method from section 4.3.
 For exercise 8 of 4.3 see the comments for excercise 11.

 Omit the material in section 4.3 on the adjoint of a matrix.
 Study the proof of property 2 of section 4.2, the "row interchange property",
 given in the text.  The proof is "sketchy" meaning, to understand it, one has
 to work through it one step at a time and fill in some of the missing steps.
 Now the text says "the proof is trivial for the case n=2".  Okay, so check it
 for the case n=2.  For the general case, draw the matrix B which is obtained
 from the matrix A by interchanging rows i and j so you can see what is going
 on in the proof.  Then write down the formula for det(B) by expanding along
 row k, a row other than row i and row j.
 
 Additional exercises:
 
 1: Let V be a vector space of dimension n and let B = {b1,...,bn} be a
    basis for V.  Let r be in R and v be in V.  Show that

            (r v)  = r (v )
                 B       B

 2: Let u=x^2+1, v=1-x^2, w=1+x+x^2 be polynomials in R[x].
    Let P2(x) = {a+bx+cx^2: a,b,c in R} denote the set of quadratic polynomials.
    Determine whether {u,v,w} is a basis for P2(x) in two ways.
    First, construct a 3 by 3 linear system of equations in r, s, t to solve
    by equating the coefficients of  1,x,x^2  in  r u + s v + t v = 0 .
    Second, compute the co-ordinate vectors uB, vB, and wB with respect to
    the basis B = (1,x,x^2) for P2(x) and solve the linear system
    r uB + s vB + t wB = 0 for r, s, t.
    Are the linear systems that you have to solve the same?

 3: Let V and W be vector spaces and T:V -> W be a linear transformation.
    Show that ker(T), the kernel of T, is a subspace of V.
 
 4: The following vector spaces are isomorphic.
    (a) U = R^(2x3) the set of 2 by 3 matrices
    (b) V = P5(x) = Span(1, x, x^2, x^3, x^4, x^5)
    (c) W = R^6
    (d) X = Span(1, x, y, xy, x^2, y^2) the quadratic polynomials in x and y
    Give an isomorphism S: U -> V.
    Give an isomorphism T: V -> W.
    Is the function F: U -> W given by F(u) = T(S(u)) an isomorphism?

 5: Consider the points (0,1), (1,1), (2,-1) in R^2.  Find the unique 
    quadratic polynomial f(x) interpolating the points, that is, satisfying
    f(0)=1, f(1)=1, f(2)=-1, using, firstly, the standard basis {1,x,x^2}
    for f(x), and secondly, using the Newton basis for f(x).