Chapter V  Vectors

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