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

The first midterm exam is on Friday, March 4, 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. 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 row equivalent, reduced row echelon form, homogeneous system of equations, and trivial solution.

1.3: Know how to perform the basic vector operations of addition and scalar multiplication algebraically and be able to interpret these geometrically. You should know all the properties listed on p. 27.

1.4: Know how to check whether a subset of ℝn is a vector subspace or not, by checking that it is nonempty, closed under vector addition and closed under scalar multiplication.

1.5: You should know what it means for a set to be linearly independent and for it to span a given subspace, and how to test whether a set fulfills these criteria (especially by using Gauss-Jordan elimination). 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.

1.6: Know how to compute the dot product of two vectors and the properties of the dot product given on p. 43. Know how to use dot products to find norms, unit vectors in a given direction, angles between vectors, and the distance between points.

1.7: Know how to fit a polynomial curve to a set of points, and use this to interpolate between them (ie, find new points on this curve). This test will not include problems on electrical network analysis or Kirchoff's Laws.

2.1 & 2.2: Know how to perform the matrix operations of addition, scalar multiplication and matrix multiplication, and their associated properties given on p. 88. 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 and main diagonal. You should know how to compute powers of matrices, and the properties given on p. 90.

2.3: Know how to find the transpose, or conjugate-transpose, of a matrix, and the properties of transpose and conjugate-transpose given on p. 100 and 106. Know what it means for a matrix to be symmetric or Hermitian, and how to prove that a matrix is symmetric or Hermitian using the definition.

2.4: 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 on p. 113 to find inverses more quickly, or show that something is an inverse of A by multiplying it by A (in both orders) to get the identity. Know how to use matrix inverses to help solve equations. Know the definition of elementary matrices and how to find them and invert them.

2.5 & 2.6: Know what we mean by transformation, domain, codomain, image, matrix transformation and linear transformation. Know how to determine whether a transformation is linear or not, and find the standard matrix of a linear transformation. You should also be able to determine whether a linear transformation is orthogonal or not. Be familiar with the standard examples of matrix transformations such as dilations/contractions, reflections and rotations (including rotation by a general angle θ). This test will not include problems using homogeneous coordinates.

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
1.1 1, 10, 11, 13
1.2 2, 6, 7
1.3 5-11
1.4 3, 4
1.5 4, 7, 12, 19
1.6 4, 8, 11, 14, 17, 23, 25
1.7 4
2.1 4, 5, 8, 10, 13
2.2 5, 7, 12, 17, 19, 25, 43
2.3 1, 3, 7, 21, 23
2.4 5, 7, 9, 16, 25, 33
2.5 5, 9, 10, 12, 14, 24
2.6 6-12, 19, 28, 29, 38