## Math 272: Linear Algebra with Applications Study Guide for the Final Exam

The final exam is on Monday, May 8, from 2:00pm to 5:00pm, in SMUD 206.

Reading week office hours are Thursday 2:00-3:30pm and Friday 1:00-3:00pm. I am also available by appointment.

Zalia Rojas at the Q-Center will hold the following drop-in hours during reading and final exam week, starting from Wednesday, May 3rd:
M-Th 11-1, 3-5
F 11-1, 3-4
Individual appointments can be scheduled through the Q-Center webpage.

The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guideline, and may not explicitly mention everything that you need to study.

The final exam is cumulative, so it is important that you study the material from earlier chapters as well.

4.4 Know how to find the matrix representing a linear transformation between vector spaces U and V, relative to given bases for each space. Make sure you can use these matrices to find the images of vectors under the transformation, and to find the kernel and range of the transformations. You should also know how to use these matrix representations to determine whether the transformation is one-to-one and/or onto.

4.5: Know what it means for matrices to be similar to each other, and how similar matrices correspond to a change of basis (as given in Theorem 15).

5.1: You must know how what it means for a vector to be an eigenvector of a matrix, and for a scalar to be an eigenvalue of a matrix. You should know what the characteristic polynomial and characteristic equation of a matrix are. You should also be very comfortable with being able to compute all the eigenvalues and corresponding eigenspaces of a matrix. This is one of the most important topics in this class, and there will certainly be a question on this on the final exam.

5.2: You must know how to diagonalize a matrix or determine that the matrix is not diagonalizable. Know how the diagonalization relates to the eigenvalues and eigenvectors of a matrix, and how it can be used to compute higher powers of the matrix.

5.4: You should know what it means for a matrix to be column stochastic, and how we know that such matrices must have 1 as an eigenvalue. You should also understand the significance of the eigenvectors corresponding to 1 in the PageRank Algorithm and in Markov processes. You should know how to interpret a Markov process statement to find the appropriate transition matrix. However, you do not need to know the formulas for the matrices used in the Google PageRank Algorithm.

6.1 & 6.2: Know how to compute the dot product of two vectors in ℝn and the properties given in Theorem 1. Know what we mean by an inner product on a vector space, and how the dot product is one example of an inner product. Given an inner product such as the dot product, you should be able to compute norms, angles, and distances using that product. You should know what we mean by an orthogonal/orthonormal set, and that such as set is linearly independent (see Theorem 5 and Corollary 1).

6.3: Know whow to obtain an orthogonal or orthonormal set from a given basis using the Gram-Schmidt Process. You should also know how to compute the projection of a vector onto another vector, or span of multiple vectors. In particular, you should know how to project a vector onto a subspace W using an orthonormal basis for W, and know that this is the closest point in W to the original vector.

6.4: This chapter includes many definitions and theorems that we have not covered in class, and therefore you may omit studying this chapter. However, you might find some useful examples relating to concepts we have already covered in Definition 1, Theorem 8.1, Proposition 5 and Theorem 10.

6.5: You do not need to memorize the least squares formula or the Fourier polynomial formulas. However, you should be able to compute with these formulas if they are given to you.

6.6:You should know that real symmetric matrices are diagonalizable and can in fact be diagonalized by an orthogonal matrix P. Know that a matrix is said to be orthogonal if its inverse equals its transpose, and that this occurs if and only if the columns of the matrix form an orthonormal set.

The departmental list of core topics for the comprehensive exams also provides a useful summary of what to review. (Note however that it does not include everything you need to study.)

Please be sure to review homework and example problems for all the chapters we have covered!

The following practice exam is intended to help you review for the exam and give you a sense of the format of the exam. It certainly does not cover all topics or question types that might appear on the exam, so please make sure you do study all topics discussed in class, included the ones covered in previous exams. However, this should provide you with a sense of the different types of questions you could encounter, including computational problems, proof problems and true/false problems.

Additionally, old Math 272 finals are available here.