We have worked extensively in the last chapter with matrices, and some with vectors. In this chapter we will develop the properties of vectors, while preparing to study vector spaces. Initially we will depart from our study of systems of linear equations, but in Section LC we will forge a connection between linear combinations and systems of linear equations in Theorem SLSLC. This connection will allow us to understand systems of linear equations at a higher level, while consequently discussing them less frequently.

Section VO Vector Operations

Subsection VEASM: Vector Equality, Addition, Scalar Multiplication

Subsection VSP: Vector Space Properties

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section LC Linear Combinations

Subsection LC: Linear Combinations

Subsection VFSS: Vector Form of Solution Sets

Subsection PSHS: Particular Solutions, Homogeneous Solutions

Subsection URREF: Uniqueness of Reduced Row-Echelon Form

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section SS Spanning Sets

Subsection SSV: Span of a Set of Vectors

Subsection SSNS: Spanning Sets of Null Spaces

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section LI Linear Independence

Subsection LISV: Linearly Independent Sets of Vectors

Subsection LINM: Linear Independence and Nonsingular Matrices

Subsection NSSLI: Null Spaces, Spans, Linear Independence

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section LDS Linear Dependence and Spans

Subsection LDSS: Linearly Dependent Sets and Spans

Subsection COV: Casting Out Vectors

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section O Orthogonality

Subsection CAV: Complex Arithmetic and Vectors

Subsection IP: Inner products

Subsection N: Norm

Subsection OV: Orthogonal Vectors

Subsection GSP: Gram-Schmidt Procedure

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection VEASM: Vector Equality, Addition, Scalar Multiplication

Subsection VSP: Vector Space Properties

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section LC Linear Combinations

Subsection LC: Linear Combinations

Subsection VFSS: Vector Form of Solution Sets

Subsection PSHS: Particular Solutions, Homogeneous Solutions

Subsection URREF: Uniqueness of Reduced Row-Echelon Form

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section SS Spanning Sets

Subsection SSV: Span of a Set of Vectors

Subsection SSNS: Spanning Sets of Null Spaces

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section LI Linear Independence

Subsection LISV: Linearly Independent Sets of Vectors

Subsection LINM: Linear Independence and Nonsingular Matrices

Subsection NSSLI: Null Spaces, Spans, Linear Independence

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section LDS Linear Dependence and Spans

Subsection LDSS: Linearly Dependent Sets and Spans

Subsection COV: Casting Out Vectors

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section O Orthogonality

Subsection CAV: Complex Arithmetic and Vectors

Subsection IP: Inner products

Subsection N: Norm

Subsection OV: Orthogonal Vectors

Subsection GSP: Gram-Schmidt Procedure

Subsection READ: Reading Questions

Subsection EXC: Exercises