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.)
  1. 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.
  2. 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.
  3. 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).
  4. 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).
  5. [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 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.