Sandra Luna McCune - Practice Makes Perfect Linear Algebra

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McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. Practice Makes Perfect: Linear Algebra is designed to help you to be successful in learning this interesting and practical subject matter. Practice Makes Perfect: Linear Algebra is designed to help you to be successful in learning this interesting and practical subject matter.

However, the book is not intended to introduce concepts, but rather its primary aim is to reinforce ideas and concepts that you have previously encountered. The topics presented are those that a competent user of linear algebra needs to know. You will find this practice study guide to be a useful supplementary text for your linear algebra course. It can also serve as a refresher text if you are using it to review previously learned linear algebra concepts and techniques. Like most topics worth knowing, learning linear algebra requires diligence and hard work. The foremost purpose of Practice Makes Perfect: Linear Algebra is as a source of solved linear algebra problems.

We believe that the best way to develop understanding, while, at the same time, acquiring accuracy and speed in linear algebra skills is to work numerous practice exercises. This book has more than 500 practice exercises from beginning to end. A variety of exercises and levels of difficulty are presented to provide reinforcement of linear algebra knowledge, understanding, and skills. In each chapter, a concept discussion followed by example problems precedes each set of exercises to serve as a concise review for readers already familiar with the topics covered. Concepts are broken into basic components to provide ample practice of fundamental skills. To use Practice Makes Perfect: Linear Algebra in the most effective way, it is important that you work through every exercise.

After working an exercise set, use the worked-out solutions to check your understanding of the concepts. We sincerely hope this book will help you acquire greater competence and confidence in using linear algebra in your future mathematical endeavors.

In this chapter, you will learn: This chapter provides some basic terminology associated with systems of linear equations and matrices and presents methods for solving linear systems. As you will see, systems of linear equations and matrices go hand-in-hand. A linear equation is an equation of the form where there are n unknowns (variables), x 1, x 2,..., x n, and a 1, a 2,..., an and c 1 are known constants. An equation is linear if all the variables occur to the first power only and if there are no products or quotients of variables.

A solution of a linear equation in n unknowns is an ordered list of n numbers s 1, s 2, ... , sn that satisfy the equation when , , ..., . In linear algebra you will need to simultaneously solve linear equations in several variables. You likely have seen two basic algebraic methods of solving simultaneous equations. An example of each method is shown for a quick review. The equation of a linear function in two variables has several forms but for the purposes of this chapter, the form is preferred.

You know from elementary algebra that the graph of such a linear equation is a line. When you encounter two linear equations, the basic question underlying a simultaneous solution is What are the coordinates of the point of intersection, if any, of the two lines that are the graphs of the two equations? This question has three possible answers. If the two lines do not intersect, there is no solution; if the two lines intersect, there is only one solution, an ordered pair ( x, y); and if the two lines are equivalent versions of the same line, then there are infinitely many solutions: an infinite set of ordered pairs. This same idea carries over to systems of m equations in n unknowns. That is,

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Given a system of m equations and n unknowns, only one of the following three possibilities for a solution to the system occurs: 1. 2. 2.

There is exactly one solution. 3. There are infinitely many solutions.

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If there is no solution to the system, the system is inconsistent. If there is at least one solution to the system, the system isNext page