As it was the case with the first two, the best way to prepare for this exam is to go through a summary (see below one or do it yourself!) of the topics discussed so far in class and read/work again the corresponding problems you solved for your homeworks. You might also want to have a look at the problems that I recommended for your individual study. Sometimes one finds later that a solution considered to be "correct" is not so correct anymore; it pays off to check your answers with the ones from the back of the textbook, with those of your colleagues or with me.

In the exam there will be problems testing both your computational and your abstract thinking skills. Let me remind you that only calculators allowing you to perform basic operations may be used. The use during the exam of a calculator that computes the reduced row echelon form or the determinant of a matrix, for instance, will be considered an act of scholastic dishonesty.

I have compiled a summary of the things we did in class after the second exam. While this exam will focus on the new material, you should realize that it is pretty comprehensive: without a good knowledge of the algorithms for solving linear systems, bringing a matrix into reduced row echelon form, checking linear independence of vectors, finding bases for linear spaces given as the kernel or as the image of a matrix, computing determinants or the Gram-Schmidt process to name just a few, without these you will not be able to do well on this exam.

The following list covers most of the topics from the book (Section 7.1 through Section 7.4 and Sections 8.1 and 8.2) that we have discussed so far. Check this list for updates after the class on Monday , December 10 as some items might be removed then if I don't have time to cover them in class.

Chapter 7

• the eigenvalues and the eigenvectors of a square matrix A
• (compute) the characteristic polynomial of A
• (solve) the characteristic equation (to find the eigenvalues)
• the algebraic multiplicity of an eigenvalue
• eigenspaces and finding bases for them
• the geometric multiplicity of an eigenvalue
• the trace of a matrix Tr(A)
• Tr(A) and det(A) as coefficients of the characteristic polynomial
• eigenbases and diagonalization of a matrix
• the case of a matrix with n different eigenvalues.
• diagonalize a matrix when possible
• compute arbitrarily large powers of diagonalizable matrices
• discrete dynamical systems given by matrices: mathematical modeling and computing their state at a given time by diagonalizing the coefficient matrix (when possible)

Chapter 8

• symmetric matrices and the spectral theorem
• orthogonally diagonalizable matrices
• compute an orthonormal eigenbasis for a symmetric matrix A.
• quadratic forms and the associated symmetric matrix
• positive definite quadratic forms and criteria for testing this property
• Updated: 12/10/007 *** ellipses and hyperbolas as level sets of quadratic forms.

The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota.