Chapter VS  Vector Spaces

We now have a computational toolkit in place and so we can begin our study of linear algebra in a more theoretical style.

Linear algebra is the study of two fundamental objects, vector spaces and linear transformations (see Chapter LT). This chapter will focus on the former. The power of mathematics is often derived from generalizing many different situations into one abstract formulation, and that is exactly what we will be doing throughout this chapter.

 Section VS Vector Spaces
  Subsection VS: Vector Spaces
  Subsection EVS: Examples of Vector Spaces
  Subsection VSP: Vector Space Properties
  Subsection RD: Recycling Definitions
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section S Subspaces
  Subsection TS: Testing Subspaces
  Subsection TSS: The Span of a Set
  Subsection SC: Subspace Constructions
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section LISS Linear Independence and Spanning Sets
  Subsection LI: Linear Independence
  Subsection SS: Spanning Sets
  Subsection VR: Vector Representation
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section B Bases
  Subsection B: Bases
  Subsection BSCV: Bases for Spans of Column Vectors
  Subsection BNM: Bases and Nonsingular Matrices
  Subsection OBC: Orthonormal Bases and Coordinates
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section D Dimension
  Subsection D: Dimension
  Subsection DVS: Dimension of Vector Spaces
  Subsection RNM: Rank and Nullity of a Matrix
  Subsection RNNM: Rank and Nullity of a Nonsingular Matrix
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section PD Properties of Dimension
  Subsection GT: Goldilocks’ Theorem
  Subsection RT: Ranks and Transposes
  Subsection DFS: Dimension of Four Subspaces
  Subsection DS: Direct Sums
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
  Annotated Acronyms VS: Vector Spaces