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 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 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 P^tAP 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 kⁿ.
- (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^tD is diagonal with every diagonal entry being either 1, -1 or 0. [The matrix Q^tD 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.]