Math 311-502 - Final Exam Review

General Information

The Final Exam will be held in our usual classroom on Friday, 12/10/04, from 12:30 to 2:30. Please bring two 8½×11 bluebook. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing linear algebra.

The test will have 8 to 10 questions, some with multiple parts. The test will directly cover material from the following sections of the book: 3.3-3.7, 14.6-8. It will also cover material from my notes listed below:

Material from chapters 1 and 2 and sections 3.1-3.2 will not be covered directly, although you will need to know it to answer questions on the material listed above. Problems will be similar to ones done for homework or as examples in class or in the notes. Material that you haven't been tested over will be weighted more heavily than old. Thus, expect sections 3.3-3.6A to be about 35%, and the rest to be about 65%, give or take a few percent. For a review of those sections, see the Test II review sheet. The other sections are dealt with below.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors. Be able to find the eigenvalues and eigenvectors for a linear transformation or matrix, and to solve simple ODE problems.
Diagonalizable transformations. A linear transformation L:V->V is said to be diagonalizable if and only if there is a basis for V relative to which the matrix for L is diagonal. Be able to determine whether or not L is diagonalizable. See my notes on Diagonalization.
Change of coordinates and similarity transformations. An n×n matrix A is similar to an n×n matrix B if there is an invertible matrix S such that B = S^-1AS. For an eigenvalue problem, a matrix A is diagonalizable if and only if it is similar to a diagonal matrix D. That is, there is an invertible matrix S such that D = S^-1AS. Here, the matrix S has a set of linearly independent eigenvectors of A for its columns. Be able to find S if A is diagonalizable.

Inner products and norms

Inner product and norm. Be able to define these terms. In simple cases, be able to verify that < , > is an inner product. Be able to state both Schwarz's inequality and the triangle inequality. Be able to find the norm of a vector and to find the angle between two vectors.
Orthogonality. Be able to define these terms: orthogonal and orthonormal sets; orthonormal bases. Know what the Gram-Schmidt procedure is and be able to use the Gram-Schmidt procedure to find an orthogonal or orthonormal set of vectors, given an inner product and a linearly independent set. Be able to find the orthogonal projection of a given, fixed vector v in V. Know the connection between this and the minimization of
d² = || v - c₁u₁ - c₂u₂ - ... - c_nu_k||²,
where B = {u₁ ... u_k} is an orthonormal set in V. Know the connection between this and the least-squares property for Fourier series.
Rotations. Know what an orthogonal matrix is and how it relates to a rotation. Given an axis of rotation and an angle, be able to find a rotation matrix. Also, be able to find the axis of rotation for a given rotation matrix.

Fourier Series and Special Functions

Fourier series. Be able to compute Fourier series for simple functions. Be able to show the orthogonality relations for the sines and cosines involved in Fourier series:
Bessel functions and Legendre polynomials. Be able to solve Bessel's equation using the method of Frobenius. Be able to prove that eigenfunctions of a self adjoint operator corresponding to distinct eigenvalues are orthogonal. Be able to use this to show that Legendre polynomials are orthogonal relative to the inner product

Updated: 12/3/04