- 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 P
_{d}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 P
_{1}. - (b) Find the orthogonal projection of x
^{2}into P_{1}.

- (a) Find an orthogonal basis for P
- 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 =
**R**^{4}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 P
^{t}AP 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
orthogonal bases.]
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 QAQ
^{t}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 I_{n}; 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 Q^{t}give an orthogonal basis of k^{n}.- (a) Demonstrate this by using row and column
operations to find an integer entry invertible matrix Q such
that QAQ
^{t}is diagonal, where A is the matrix given in the previous problem, and k =**R**. - (b) Verify that the columns of Q
^{t}give an orthogonal basis. - (c) Find an invertible diagonal matrix D such that
DQAQ
^{t}D is diagonal with every diagonal entry being either 1, -1 or 0. [The matrix Q^{t}D 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.]

- (a) Demonstrate this by using row and column
operations to find an integer entry invertible matrix Q such
that QAQ