Chapter M  Matrices

We have made frequent use of matrices for solving systems of equations, and we have begun to investigate a few of their properties, such as the null space and nonsingularity. In this chapter, we will take a more systematic approach to the study of matrices.

 Section MO Matrix Operations
  Subsection MEASM: Matrix Equality, Addition, Scalar Multiplication
  Subsection VSP: Vector Space Properties
  Subsection TSM: Transposes and Symmetric Matrices
  Subsection MCC: Matrices and Complex Conjugation
  Subsection AM: Adjoint of a Matrix
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section MM Matrix Multiplication
  Subsection MVP: Matrix-Vector Product
  Subsection MM: Matrix Multiplication
  Subsection MMEE: Matrix Multiplication, Entry-by-Entry
  Subsection PMM: Properties of Matrix Multiplication
  Subsection HM: Hermitian Matrices
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section MISLE Matrix Inverses and Systems of Linear Equations
  Subsection IM: Inverse of a Matrix
  Subsection CIM: Computing the Inverse of a Matrix
  Subsection PMI: Properties of Matrix Inverses
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section MINM Matrix Inverses and Nonsingular Matrices
  Subsection NMI: Nonsingular Matrices are Invertible
  Subsection UM: Unitary Matrices
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section CRS Column and Row Spaces
  Subsection CSSE: Column Spaces and Systems of Equations
  Subsection CSSOC: Column Space Spanned by Original Columns
  Subsection CSNM: Column Space of a Nonsingular Matrix
  Subsection RSM: Row Space of a Matrix
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section FS Four Subsets
  Subsection LNS: Left Null Space
  Subsection CRS: Computing Column Spaces
  Subsection EEF: Extended echelon form
  Subsection FS: Four Subsets
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions