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

The final exam is on Tuesday, May 10, from 9:00am to 12:00pm, in SMUD 206.

Reading period office hours are Monday 4:00-5:00pm. Cole Hawkins will also hold his usual office hours Sunday 7:30-8:30pm in SMUD 205.

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.

4.8: Know what we mean by the kernel and range of a transformation and how to find them. Make sure you know the statements of the theorems in this chapter, including the Rank-Nullity Theorem applied to transformations.

4.9: Know what it means for a transformation to be one-to-one and onto and how to prove these properties in general, but especially for linear transformations. You should also know how to determine whether a transformation is invertible or not and be able to find the inverse for a linear transformation using matrix representations and inverse matrices. You should also know that an isomorphism of vector spaces is a transformation that is one-to-one, onto and linear.

4.10: You should be able to find the full set of solutions for a system of nonhomogeneous linear equations.

5.1 Given a basis of a general vector space, you should be able to find the coordinate vector for a given vector in that space. You must also know how to find the transition matrix between two bases and how to use this to find the coordinate vectors in different bases. Know that the transformation taking a vector in an n-dimensional vector space U to its coordinate vector relative to a given basis is an isomorphism from U to ℝn.

5.2 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 be able to use these matrices to find the kernel and range of the transformations. You should also 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 6).

5.3: 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. You are not required to know the subsections in 5.3 on symmetric matrices and orthogonal diagonalization.

4.6: Know what we mean by an orthogonal/orthonormal set, and how to obtain one from a given basis using the Gram-Schmidt Process. You should also know the results in this chapter relating to orthogonal matrices, and know how to compute the projection of a vector onto another vector. You do not need to know about QR factorizations.

4.7: You should know what we mean by the orthogonal complement of a vector space, and how to find the orthogonal decomposition of a vector relative to a subspace W using Theorems 26 and 27 in this section. 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.

6.1: Know what we mean by an inner product on a vector space, and how the dot product is one example of an inner product. In particular, you should know how to check that a given function on pairs of vectors is an inner product by showing that it satisfies the 4 conditions of the definition. Given an inner product, you should be able to compute norms, angles, and distances using that product.

The departmental list of core topics for the comprehensive exams also provides a useful summary of what to review.

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 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.

Below is a list of relevant additional practice problems for each section.

Section Suggested Problems
4.8 3a, 4, 11, 13, 16, 22, 29, 41
4.9 1, 4, 6
5.1 5, 6, 10, 12, 13, 23, 25, 27
5.2 8, 9c, 11, 17, 19, 20, 22b
5.3 3, 4, 9, 10, 13, 20
4.6 1-4, 14-16, 22-23
4.7 2, 4, 6, 7
6.1 1, 4, 6, 8a, 9a, 10a, 12, 18a, 28acd