## Math 272: Linear Algebra with Applications Study Guide for Exam 2

The second midterm exam is on Friday, April 11, during our regular class time.

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. Although this exam will emphasize chapters covered since the first midterm, it is still very important to know the earlier material as later chapters build on this material. The information here is only intended to serve as a guideline, and may not explicitly mention everything that you need to study.

3.1 & 3.2: Know how to compute the determinant of a matrix using cofactors, by expanding along a convenient row or column. You should also know the effect of each elementary row operation on the determinant of a matrix, and be able to use this to compute a determinant via Gauss-Jordan elimination. Make sure you know all of the properties of the determinant given in Section 3.2.

3.3: You do not need to know about adjoint matrices or Cramer's rule, but you should still remember how to find the inverse of a matrix using the methods of Section 2.4. You should also know all the equivalent statements to the invertibility of a matrix, as listed in the class handout.

3.4: 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 also be very comfortable with being able to compute all the eigenvalues and corresponding eigenspaces of a matrix.

3.5: 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. However, you do not need to know the formulas for the matrices used in the Google PageRank Algorithm.

4.1: You should know what we mean by an abstract vector space, which is defined by the 10 axioms given on p. 216. You should also know what we mean by a subspace, and how to check that a subset of a given vector is a subspace by only verifying axioms 1 and 2. You may also use the fact that a subset does not contain the zero vector to quickly determine that it is not a subspace.

4.2: Know what we mean by a linear combination of vectors, and what is means for vectors to span a vector space. Know how to find a spanning set for a given space, and how to determine whether a vector is in the span of a given set or not.

4.3: Know what it means for a set of vectors to be linearly independent or linearly dependent, and how to determine which one a given set of vectors is.

4.4: You should know that a basis is a linearly independent set that spans a space. You should also know how to determine whether a given set is a basis of a subspace, and how to find a basis of a subspace if none is given. Know how to use the basis to determine the dimension of the subspace.

4.5: Know how to determine the rank of a matrix, and find its row space, column space and null space by finding a basis for each of these. Know the statement of the Rank-Nullity Theorem.

Please be sure to review homework and example problems for the chapters given above!

The following two practice exams are intended to help you review for the exam and give you a sense of the format of the exam. They certainly do not cover all topics that might appear on the exam, so please make sure you do study all topics discussed in class. 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.

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

Section Suggested Problems
3.1 2, 6, 11, 12
3.2 6, 8, 9, 10, 14, 24, 25
3.3 1, 3, 5, 13, 27
3.4 1-14, 23-31
3.5 1, 8
4.1 1, 3, 7, 16, 20, 21, 27, 32, 34
4.2 2, 8, 9, 12, 15, 16, 19
4.3 1, 6, 8, 9, 15, 16
4.4 1, 4, 6, 15, 17, 19, 20, 33
4.5 1, 2, 3, 6, 12, 14, 19

Maintained by ynaqvi and last modified 04/08/16