Math 817: Problem set 6
Instructions: Write up any four of the following problems (but
you should try all of them, and
in the end you should be sure you know how to do all of them).
Your write ups are due Friday, October 8, 2004.
Each problem is worth 10 points, 9 points for correctness
and 1 point for communication. (Your goal is not only
to give correct answers but to communicate
your ideas well. Make sure you use good English,
so proofread your solutions.
Once you finish a solution, you
should restructure awkward sentences, and strike out
anything that is not needed in your approach to the problem.)

 (a) Let M be the matrix whose columns are
(1, 2, 1, 0)^{t},
(4, 8, 4, 3)^{t},
(0, 1, 3, 4)^{t}, and
(2, 5, 1, 4)^{t}. Find a basis for the
subspace of R^{4} spanned by the columns of M
(the socalled column space of M).
 (b) Let A be the matrix whose rows are
(2, 1, 2, 3), and
(1, 1, 3, 0). The set of solutions
x to Ax = 0 is a subspace of
R^{4} called the nullspace of A.
Find a basis for the nullspace of A.
 Let n be a positive integer, let
F be a field and let W be a subspace of F^{n}.
Let W' be the set of all v in F^{n}
such that v^{t}w is 0 for all
w in W.
 (a) Show that W' is a subspace of F^{n}.
 (b) Show that W'' contains W.
 The set of all symmetric real nxn matrices
is a subspace W of the real vector space M_{n}(R)
of all real nxn matrices.
Exhibit a basis and determine the dimension of W.
 Problem #5 from 3.4:
 (i) Prove that the set B =
{(1,2,0)^{t}, (2,1,2)^{t}, (3,1,1)^{t}}
is a basis of R^{3}.
 (ii) Find the coordinate vector of the vector
v = (1,2,3)^{t} with respect to B
(ordered as given).
 (iii) Let B' =
{(0,1,0)^{t}, (1,0,1)^{t}, (2,1,0)^{t}}.
Find the change of basis matrix P relating the old basis
B to the new basis B'.
 (iv) For which primes p is B a basis of
F_{p}^{3}?
 Problem #11 from 3.4: Let F = F_{p}.
 (a) Compute the order of SL_{2}(F).
 (b) Compute the number of (ordered) bases of F^{n},
and the orders of GL_{n}(F) and SL_{n}(F).