Math 817: Problem set 7
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 15, 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.)
- First, some prologue: An mxn matrix A with entries in a field F gives a
linear transformation TA: Fn -> Fm,
by matrix multiplication.
The rank of A is defined to be the rank of TA.
(Since rank(TA) = dim Im(TA), we see
that the rank of A is just the dimension of the
column space of A, which is given by counting the number of pivots
in the row reduced matrix for A, which itself is just the matrix
for TA, but with respect to a different basis
for Fm.) Now the problem.
Show that A and At,
the transpose of A (p. 18 of Artin), have the same rank.
- Let F be a field and let W be a subspace of Fm.
Recall from the previous homework the subspace W'.
- (a) Prove that dim W + dim W' = dim Fm.
- (b) Prove that W = (W')'.
- (c) Let i be an integer between 0 and m,
inclusive. Prove that there is a bijection between subspaces
of Fm of dimension i and those of dimension m-i.
- Let A be an mxn matrix with entries in a field F,
and let TA: Fn -> Fm be
the corresponding linear transformation.
Let V be a subspace of Fn, and let W be a subspace
of Fm such that TA(V) is contained in W.
Let T' be the linear transformation corresponding to
At. Show that T'(W') is contained in V', where
the operation ' is defined as on the previous homework.
- Count the number of subspaces of each dimension in F3
if F is the field F5. Justify your answers.
- (a) Find the rank of the matrix given in problem 1.2 of
Artin, p. 145.
- (b) When F is a finite field with p elements (p a prime),
formula 1.6 (p. 110) and formula 6.15 (p. 58) both apply
to a linear transformation of finite dimensional vector spaces over F.
Show how each formula implies the other.
(Once you understand what to do, this one is easy!)
- (Problem 2.9 on p. 146 of Artin.) Let S and T be linear transformations
from V to W. Define S+T and cT by the rules [S+T](v) = S(v) + T(v)
and [cT](v) = c(T(v)).
- (a) Prove that S+T and cT are linear transformations, and
describe their matrices in terms of the matrices for S and T.
- (b) Let L be the set of all linear transformations
from V to W. Prove that these laws make L into a vector space,
and compute its dimension.
- (Problem 3.5 on p. 146 of Artin.) Find all of the invariant
subspaces of the real linear operators TA and TB
whose associated matrices are (a) A and (b) B, where
A is 2x2, has a 0 at bottom left, and otherwise its entries are 1s,
and B is a 3x3 diagonal matrix with 1, 2 and 3 on the diagonal.
(I.e., find each invariant subspace, and justify why
there are no others.)