Math 818: Problem set 1, due January 21
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, January 21, 2005.
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.)
- Let V be a real n dimensional vector space with a
symmetric positive definite bilinear form < , >.
For each v in V, define |v| = (< v,v >)1/2.
Show how to use the Schwarz inequality to prove the
triangle inequality, that |v+w| <= |v| + |w|, for
all v and w in V.
- Let V be the real vector space of all
real polynomials f(x).
Let Pd be the subspace spanned by polynomials
of degree at most d. Define < f,g > by integrating
f(x)g(x) from 0 to 1 with respect to x; this defines
a real symmetric bilinear form on V.
- (a) Find an orthogonal basis for P1.
- (b) Find the orthogonal projection of x2 into P1.
- Let W be the real vector space of all continuous real valued functions
from the reals to the reals. Let V be the
3 dimensional subspace of W spanned by f(x) = 1, g(x) = x, and h(x) = |x|.
Define a real symmetric bilinear form on V by
taking < v,w > to be the integral of
v(x)w(x) from 0 to 1 with respect to x (thus < f,g > is,
for example, the integral of f(x)g(x) = x from 0 to 1, which is 1).
- (a) Find an orthogonal basis for V.
- (b) What is the signature of < , >? I.e., find p, n and z.
- (c) Repeat (a) and (b) if we instead define
< v,w > to be v(1)w(1) (thus < f,g > = f(1)g(1) = 1).
[Note: both definitions of
< , > are in terms of integrals, Riemann-Stieltjes integrals.
For both definitions we're integrating a function from 0 to 1,
but with respect to dx for the first definition, and with respect to da
in the second, where the function a(x)
is defined to be 0 for x < 1, and to be 1 for x >= 1.]
- Let V = R4 be the standard 4 dimensional
real vector space, equipped with the bilinear form < , >A
where A is the matrix whose rows, in order, are: (0 3 0 3), (3 0 3 2), (0 3 0 3) and
(3 2 3 0).
- (a) Find an orthogonal basis for V. Explain how you found it.
(It can be helpful to choose a basis consisting of vectors with integer
entries, when possible. Here it is possible.)
- (b) What is the signature of the bilinear form?
- (c) Find an invertible matrix P such that PtAP is a diagonal
matrix, and such that every diagonal entry is either -1, 0 or 1.
(You can in this case express P in the form QD, where Q is a 4x4 integer
matrix, and D is an invertible diagonal matrix.)
- [This problem gives an alternate method of finding
If A is an nxn symmetric matrix with entries in a field k
of characteristic not 2,
then there is an invertible matrix Q with entries in k
such that QAQt is diagonal. [Apply a
sequence of row operations to clean up column 1 of A,
giving a matrix A' with zeros below the diagonal
in the first column; then, since A is symmetric, the same
operations but applied to the columns of A'
clean up row 1, giving a symmetric matrix A'' with zeros in
the first row and first column except for the entry on the diagonal.
Now do row and column 2, etc. Eventually you get a diagonal matrix.
To get the matrix Q,
apply the same sequence of row operations to In; the result is
Q.] If the entries of A are integers
and k is the reals or rationals, we can even assume Q has integer
entries. [At the end, just multiply Q by an integer which wipes out
any denominators. Alternatively, never use a row operation
involving division. The Q you end up with then
has only integer entries; you won't have to wipe out
denominators.] The columns of Qt give an orthogonal
basis of kn.
- (a) Demonstrate this by using row and column
operations to find an integer entry invertible matrix Q such
that QAQt is diagonal, where A is the matrix given in the
previous problem, and k = R.
- (b) Verify that the columns of Qt give an orthogonal
- (c) Find an invertible diagonal matrix D such that
DQAQtD is diagonal with every diagonal entry being either 1, -1 or 0.
[The matrix QtD gives one possible answer for P in part (c)
of the previous problem. Since P is not unique, your answer
here may be different from your answer above.]