## Math 272: Linear Algebra with Applications Study Guide for Exam 1

The first midterm exam is on Wednesday, March 1, during our regular class period.

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.

1.1 & 1.2: It is very important that you know how to use Gauss-Jordan Elimination! You should be comfortable using this to solve systems of linear equations, and determining the number of solutions that a system of equations has. Know what we mean by the terms free variable, augmented matrix, row equivalent, row echelon form, and reduced row echelon form.

1.3: Know how to perform the matrix operations given in this chapter, including addition, scalar multiplication, matrix multiplication and transposing. You should know all of the properties listed in this chapter, given in Theorems 4,5 and 6. Pay careful attention to the fact that the order matters in matrix multiplication and that equations involving matrix multiplication do not allow for cancellation in general! You should know what we mean by the terms zero matrix, identity matrix, square matrix, main diagonal and symmetric matrix.

1.4: You should know what it means for a matrix to be invertible (and what we mean by inverse of a matrix). Be able to compute the inverse of a matrix or show that it does not exist. You should also be able to use the properties of matrix inverse given in class, and show that something is an inverse of A by multiplying it by A to get the identity.

1.5: Know how to use matrix inverses to help solve equations, and that invertibility of the coefficient matrix implies that you have a unique solution for your linear system. Know what we mean my a homogeneous linear system and by trivial and nontrivial solutions.

1.6: Know how to compute the determinant of a matrix. You should be able to find it by direct calculation for 2 by 2 and 3 by 3 matrices, and all upper triangular matrices. You should also know how to use row operations to find the determinant by reducing to an upper triangular matrix. You should know all the properties of determinants given in this chapter (see Theorems 14-17 and Corollary 1). You do not need to know Cramer's rule.

1.7: Know the definition of elementary matrices and how to find them and invert them. You should be able to write an invertible matrix as a product of elementary matrices. You should also know how to find the LU Decomposition of a matrix and be able to use this decomposition to solve systems of equations. You do not need to know how to find PLU Decompositions.

2.1: Know the notation for Euclidean n-space, and what we mean by it. Know how to interpret vector operations algebraically and geometrically, and know the properties given in Theorem 1.

2.2 & 2.3: You must know the definition of a linear combination, and be able to determine (and prove) whether a vector is a linear combination of a set of vectors or not. You must also know the definitions of linear independence and linear dependence, and how to prove whether a set of vectors is linearly independent or not. Finally, you should know the theorems discussed in class that associate these concepts with systems of equations, pivot positions in matrices and the invertibility of matrices.

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

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! 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
1.1 11, 19, 25, 29, 33, 39
1.2 21-28, 31, 39, 45, 51, 52
1.3 5, 9, 17, 21, 32, 35, 41
1.4 5, 9, 23, 31, 33, 40
1.5 13, 17, 19, 23, 25, 27
1.6 2, 3, 11, 13, 19, 21, 27-30, 35, 39, 41, 52
1.7 1-4, 5, 7, 11, 13, 17, 19
CT1 1-45
2.1 5, 15, 17, 21, 27
2.2 5, 7, 13, 17, 25, 31, 33, 35
2.3 7, 11, 15-18, 19, 24, 27, 37, 39
CT2 1-33

Maintained by ynaqvi and last modified 02/26/17