We will motivate our study of linear algebra by studying solutions to systems of linear equations. While the focus of this chapter is on the practical matter of how to find, and describe, these solutions, we will also be setting ourselves up for more theoretical ideas that will appear later.

Section WILA What is Linear Algebra?

Subsection LA: “Linear” + “Algebra”

Subsection AA: An Application

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section SSLE Solving Systems of Linear Equations

Subsection SLE: Systems of Linear Equations

Subsection PSS: Possibilities for Solution Sets

Subsection ESEO: Equivalent Systems and Equation Operations

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section RREF Reduced Row-Echelon Form

Subsection MVNSE: Matrix and Vector Notation for Systems of Equations

Subsection RO: Row Operations

Subsection RREF: Reduced Row-Echelon Form

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section TSS Types of Solution Sets

Subsection CS: Consistent Systems

Subsection FV: Free Variables

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section HSE Homogeneous Systems of Equations

Subsection SHS: Solutions of Homogeneous Systems

Subsection NSM: Null Space of a Matrix

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section NM Nonsingular Matrices

Subsection NM: Nonsingular Matrices

Subsection NSNM: Null Space of a Nonsingular Matrix

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Subsection LA: “Linear” + “Algebra”

Subsection AA: An Application

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section SSLE Solving Systems of Linear Equations

Subsection SLE: Systems of Linear Equations

Subsection PSS: Possibilities for Solution Sets

Subsection ESEO: Equivalent Systems and Equation Operations

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section RREF Reduced Row-Echelon Form

Subsection MVNSE: Matrix and Vector Notation for Systems of Equations

Subsection RO: Row Operations

Subsection RREF: Reduced Row-Echelon Form

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section TSS Types of Solution Sets

Subsection CS: Consistent Systems

Subsection FV: Free Variables

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section HSE Homogeneous Systems of Equations

Subsection SHS: Solutions of Homogeneous Systems

Subsection NSM: Null Space of a Matrix

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section NM Nonsingular Matrices

Subsection NM: Nonsingular Matrices

Subsection NSNM: Null Space of a Nonsingular Matrix

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions