# Linear Algebra and its Applications, Global Edition (5e)

##### Description

*For courses in linear algebra.*

With traditional linear algebra texts, the course is relatively easy for students during the early stages as material is presented in a familiar, concrete setting. However, when abstract concepts are introduced, students often hit a wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations) are not easily understood and require time to assimilate. These concepts are fundamental to the study of linear algebra, so students' understanding of them is vital to mastering the subject. **This text** makes these concepts more accessible by introducing them early in a familiar, concrete *R ^{n}* setting, developing them gradually, and returning to them throughout the text so that when they are discussed in the abstract, students are readily able to understand.

##### Table of contents

**1. Linear Equations in Linear Algebra**- Introductory Example: Linear Models in Economics and Engineering
- 1.1 Systems of Linear Equations
- 1.2 Row Reduction and Echelon Forms
- 1.3 Vector Equations
- 1.4 The Matrix Equation Ax = b
- 1.5 Solution Sets of Linear Systems
- 1.6 Applications of Linear Systems
- 1.7 Linear Independence
- 1.8 Introduction to Linear Transformations
- 1.9 The Matrix of a Linear Transformation
- 1.10 Linear Models in Business, Science, and Engineering
**2. Matrix Algebra**- Introductory Example: Computer Models in Aircraft Design
- 2.1 Matrix Operations
- 2.2 The Inverse of a Matrix
- 2.3 Characterizations of Invertible Matrices
- 2.4 Partitioned Matrices
- 2.5 Matrix Factorizations
- 2.6 The Leontief Input–Output Model
- 2.7 Applications to Computer Graphics
- 2.8 Subspaces of Rn
- 2.9 Dimension and Rank
**3. Determinants**- Introductory Example: Random Paths and Distortion
- 3.1 Introduction to Determinants
- 3.2 Properties of Determinants
- 3.3 Cramer’s Rule, Volume, and Linear Transformations
**4. Vector Spaces**- Introductory Example: Space Flight and Control Systems
- 4.1 Vector Spaces and Subspaces
- 4.2 Null Spaces, Column Spaces, and Linear Transformations
- 4.3 Linearly Independent Sets; Bases
- 4.4 Coordinate Systems
- 4.5 The Dimension of a Vector Space
- 4.6 Rank
- 4.7 Change of Basis
- 4.8 Applications to Difference Equations
- 4.9 Applications to Markov Chains
**5. Eigenvalues and Eigenvectors**- Introductory Example: Dynamical Systems and Spotted Owls
- 5.1 Eigenvectors and Eigenvalues
- 5.2 The Characteristic Equation
- 5.3 Diagonalization
- 5.4 Eigenvectors and Linear Transformations
- 5.5 Complex Eigenvalues
- 5.6 Discrete Dynamical Systems
- 5.7 Applications to Differential Equations
- 5.8 Iterative Estimates for Eigenvalues
**6. Orthogonality and Least Squares**- Introductory Example: The North American Datum and GPS Navigation
- 6.1 Inner Product, Length, and Orthogonality
- 6.2 Orthogonal Sets
- 6.3 Orthogonal Projections
- 6.4 The Gram–Schmidt Process
- 6.5 Least-Squares Problems
- 6.6 Applications to Linear Models
- 6.7 Inner Product Spaces
- 6.8 Applications of Inner Product Spaces
**7. Symmetric Matrices and Quadratic Forms**- Introductory Example: Multichannel Image Processing
- 7.1 Diagonalization of Symmetric Matrices
- 7.2 Quadratic Forms
- 7.3 Constrained Optimization
- 7.4 The Singular Value Decomposition
- 7.5 Applications to Image Processing and Statistics
**8. The Geometry of Vector Spaces**- Introductory Example: The Platonic Solids
- 8.1 Affine Combinations
- 8.2 Affine Independence
- 8.3 Convex Combinations
- 8.4 Hyperplanes
- 8.5 Polytopes
- 8.6 Curves and Surfaces
**Appendices**- A. Uniqueness of the Reduced Echelon Form
- B. Complex Numbers

##### New to this edition

**New to the textbook**

- More than 25% of the
**exercises are new or updated**, especially computational exercises. These are crafted in a way that reflects the substance of each of the sections they follow, developing the students’ confidence while challenging them to practice and generalize the new ideas they have encountered. **Conceptual Practice Problems**and their solutions in most sections provide additional support for proof- or concept-based learning. Additional guidance has also been added to some of the proofs of theorems in the body of the text.

**MyMathLab not included. **Students, if MyMathLab is a recommended/mandatory component of the course, please ask your instructor for the correct ISBN and course ID. MyMathLab should only be purchased when required by an instructor. Instructors, contact your Pearson representative for more information.

**Also available with MyMathLab**

MyMathLab is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. MyMathLab includes assignable algorithmic exercises, the complete eBook, interactive figures, tools to personalize learning, and more.

**More assignable algorithmic exercises**allow you to craft homework assignments that better meet student needs.**Interactive eBook**utilizes Wolfram CDF Player (the free Mathematica player). Students can interact with figures and experiment with matrices by looking at numerous examples.**Interactive figures**bring the geometry of linear algebra to life. Using the Wolfram CDF Player, students can manipulate figures and experiment with matrices by looking at numerous examples. These figures are available within the eBook and as separate files (for ease of use during lecture).**Technology exercises and projects**in MATLAB, Maple, Mathematica, and TI format have been updated to reflect changes in those systems. And all resources have been reorganized to make them easier for instructors and students to locate and use.

##### Features & benefits

**Early introduction of key concepts:**Fundamental ideas of linear algebra are introduced within the first seven lectures, in the concrete setting of*R*, then gradually examined from different points of view. Later, generalisations of these concepts appear as natural extensions of familiar ideas.^{n}**Linear transformations**form a “thread” that is woven into the fabric of the text. Their use enhances the geometric flavor of the text. In Chapter 1, for instance, linear transformations provide a dynamic and graphical view of matrix-vector multiplication.**Orthogonality and Least-Squares Problems**receive more comprehensive treatments than is commonly found in beginning texts because orthogonality plays such an important role in computer calculations and numerical linear algebra and because inconsistent linear systems arise so often in practical work.**Eigenvalues appear fairly early in the text**, in Chapters 5 and 7. Because this material is spread over several weeks, students have more time to absorb and to review these critical concepts. Eigenvalues are motivated by and applied to discrete and continuous dynamical systems, which appear in Sections 1.10, 4.8, and 4.9, and in five sections of Chapter 5.**A modern view of matrix multiplication**is presented, with definitions and proofs focusing on the columns of a matrix rather than on the matrix entries.**Focus on visualisation**of concepts throughout the book helps students grasp the concepts.Each major concept in the course is given a geometric interpretation because many students learn better when they can visualise an idea.**Numerical Notes**provide a realistic slant to the text. Students are reminded frequently of issues that arise in real-life applications of linear algebra.**Applications**are varied and relevant. Some applications appear in their own sections; others are treated within examples and exercises. Each chapter opens with an introductory vignette that sets the state for some applications of linear algebra and provides a motivation for developing the mathematics that follows.**Exercise sets are meticulously constructed**and consist of the following elements. Each section features an abundant supply of exercises, ranging from routine computations to conceptual questions to applications. Innovative questions pinpoint conceptual difficulties that the authors have found in student papers over the years.- A few carefully selected
**Practice Problems**appear just before each exercise set. Complete solutions follow the exercise set. These problems either focus on potential trouble spots in the exercise set or provide a “warm-up” for the exercises, and the solutions often contain helpful hints or warnings about the homework. **True/False Questions**appear just after the computational exercises and encourage students to read the text and think critically.**NEW! Conceptual Practice Problems**and their solutions in most sections provide proof- or concept-based examples for students to review.**[M] exercises**appear in every section. To be solved with the aid of a [M]atrix program such as MATLAB^{™}, Maple^{®}, Mathematica^{®}, MathCad^{®}, Derive^{®}or programmable calculators with matrix capabilities, such as the TI-83 Plus^{®}, TI-86^{®}, TI-89^{®}, and HP-48G^{®}. Data for these exercises are provided on the Web.

- A few carefully selected