MAC 2313



MAS 3105

Review - Exam 1

September 16, 2010

When: September 23, 2010, in class

What Material: Sections 1.1-1.4, 1.6-1.7, 2.1, 2.3-2.4, 2.7-2.8.

Procedure: The exam will be closed book. You will be allowed one sheet of notes, 8-1/2 by 11 inches, front and back. You may not use any graphing calculators of any kind. You may bring in a basic calculator if you want.

How to Study: The exam will have straightforward computation questions (find the inverse of a matrix, solve a system of equations, determine if a vector is in the span of a set, etc) and the exam will have “understanding” questions. These are questions that test whether or not you understand the definitions and theorems from the class. All the problems from the homework that were at the end of a section – those were “understanding” questions.

You WILL be expected to show your work on all problems. That means you need to show all steps in a Gaussian elimination.

You should run through all of the True/False questions. These really help you with definitions and theorems.

Computations are straightforward so far, and everything boils down to finding linear combinations and/or performing Gaussian elimination. Make sure you can do both tasks quickly, and make sure you don’t make arithmetic mistakes.

You should go over every homework problem assigned, as well as all problems from the chapters that are similar. Continue to practice until you can do the problems comfortably in a test situation (one page of notes and a calculator, no additional notes, no back of the book, no help from friends). Also focus on quiz problems. If I found a type of problem important enough for a quiz, it is probably important enough for an exam. Finally go over the review exercise sets at the end of the chapter. Even though it includes some material that I did not cover, it is good in that the topics are mixed together, just like on the exam.

Specific Topics:

• Matrix and Vector Arithmetic: You need to be able to add, subtract, and scalar multiply both matrices and vectors. You need to be able to do matrix-vector multiplication, matrix-matrix multiplication, as well as vector linear combinations. Make sure you know all of the algebra rules (Theorems 1.1, 1.3, 2.1). You need to be able to transpose a matrix, and you need to understand the definitions associated with transposes (symmetric matrices and skew-symmetric matrices). Also know how transpose and arithmetic interact (Theorems 1.2 and 2.1-g). Computation questions: 1.1: 1-24, 1.2: 1-16, 2.1: 1-32. Understanding questions: 1.1: 57-80, 1.2: 80-89, 2.1: 54-57, 59-64, 67, 68, 70.

• Systems of Linear Equations: Given a system of linear equations, determine if the system is consistent. If so, find the solution set. For this (as well as many other problems), you need to find the reduced row echelon form of a matrix, and you will do that using Gaussian elimination. Computation questions: 1.3: 1-54, 1.4: 1-42 (17-34 especially). Understanding \questions: 1.3: 78-86, 1.4: 73-93.

• Span: Given a set of vectors, determine if another vector is in the span of the set. Also, given a set of vectors, determine if they generate all of Rm. In both cases, this requires Gaussian elimination. Make sure you know Theorem 1.6 – it can save you a bunch of work. Computation questions: 1.6: 1-36 (17-20 especially). Understanding questions: 1.6: 65-76.

• Linear Independence: Given a set of vectors, determine if the set is linearly dependent or linearly independent. Again, it all boils down to Gaussian elimination. And again, make sure you know Theorem 1.8. Computation questions: 1.7: 1-50 (especially 39-50). Understanding questions: 1.7: 83-89.

• Invertible Matrices: Given a square matrix, find its inverse, or determine if no inverse exists. Given an elementary row operation, find the corresponding elementary matrix. Make sure you know how inverses interact with other things we’ve learned (Theorem 2.2), and make sure you know all of the equivalent conditions for a matrix to be invertible (Theorem 2.6). Computation questions: 2.3: 9-32, 2.4: 1-18. Understanding questions: 2.3: 56, 58-65, 2.4: 64-73, 89-91.

• Linear Transformations: Given a linear transformation, write down its standard matrix, determine if it is one-to-one, determine if it is onto, and find its nullspace. Given some information about a linear transformation (like the image of a couple of vectors), determine the linear transformation. Make sure you know how linear transformations interact with linear combinations (Theorems 2.7 and 2.8), and make sure you know how to tell if a map is one-to-one or onto (Theorems 2.10 and 2.11). Computation questions: 2.7: 1-34, 56-71, 2.8: 1-40, 63-68. Understanding questions: 2.7: 87-100.

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