Elementary Linear Algebra

• Italic: Definition
• Bold: Theorem/Corollary

Chapters

Chapter 1: Systems of Linear Equations and Matrices

• 1.1 Introduction to Systems of Linear Equations
  • Linear Equation
  • Homogeneous Linear Equation
  • Linear System
  • Solution of Linear System
  • Consistency of Linear System
  • Augmented Matrix
  • Elementary Row Operation
• 1.2 Gaussian Elimination
  • Row Echelon Form (REF)
  • Reduced Row Echelon Form (RREF)
  • General Solution of Linear System
  • Gauss-Jordan Elimination
  • Gaussian Elimination
  • Homogeneous Linear Equation
  • Trivial Solution
• 1.3 Matrices and Matrix Operations
  • Matrix
  • Matrix Equality
  • Matrix Operations
  • Matrix Scalar Multiplication
  • Matrix Product
  • Partitioned Matrices and Submatrix
  • Matrix Multiplication by Columns and by Rows
  • Product of Partitioned Matrices
  • Linear Combination of Matrices
  • Transpose Matrix
  • Trace of a Matrix
• 1.4 Inverses; Algebraic Properties of Matrices
  • Properties of Matrix Arithmetic
  • Zero Matrix
  • Properties of Zero Matrix
  • Identity Matrix
  • Form of RREF
  • Inverse Matrix
  • Properties of Inverse Matrix
  • Matrix Invertibility
  • Inverse of Matrix Product
  • Inverse of 2x2 Matrix
  • Powers of a Matrix
  • Properties of Negative Exponents of Matrix
  • Properties of Transpose Matrix
  • Transpose of Invertible Matrix
• 1.5 Elementary Matrices and a Method for Finding A^1
  • Row Equivalent
  • Elementary Matrix
  • Row Operations by Matrix Multiplications
  • Invertibility of Elementary Matrix
  • Inversion Algorithm
• 1.6 More on Linear Systems and Invertible Matrices
  • Number of Solutions of a Linear System
  • Solution of Linear System by Matrix Inversion
  • Properties of Invertible Matrices
• 1.7 Diagonal, Triangular, and Symmetric Matrices
• 1.8 Matrix Transformations

Chapter 2: Determinants

• 2.1 Determinants by Cofactor Expansion
  • Determinant of Matrices
• 2.2 Evaluating Determinants by Row Reduction
• 2.3 Properties of Determinants; Cramer’s Rule
  • Matrix Equivalency Statements
  • Determinant of Matrix Product

Chapter 3: Euclidean Vector Spaces

• 3.1 Vectors in 2-Space, 3-Space, and n-Space
  • Ordered n-Tuple
  • n-Space
  • Zero Vector
  • Vector Equality
  • Vector Operations
  • Properties of Vector Operations
  • Linear Combination
• 3.2 Norm, Dot Product, and Distance in Rn
  • Norm
  • Properties of Norm
  • Unit Vector
  • Distance of Vectors
  • Dot Product
  • Algebraic Properties of Dot Product
  • Cauchy-Schwarz Inequality
  • Triangle Inequality for Vectors and Distances
  • Parallelogram Equation for Vectors
  • Relationship between Dot Product and Norm in Rn
  • Dot Products as Matrix Multiplication
• 3.3 Orthogonality
  • Orthogonal
  • Normal
  • Point-Normal
  • Orthogonal Projection
• 3.4 The Geometry of Linear Systems
• 3.5 Cross Products
  • Cross Product

Chapter 4: General Vector Spaces

• 4.1 Real Vector Spaces
  • Vector Space
• 4.2 Subspaces
  • Subspace
  • Intersection of Subspaces
  • Span
  • Solution Space
  • Consistency of Linear System by its Constant Vector
• 4.3 Linear Independence
  • Linear Independence
• 4.4 Coordinates and Basis
  • Basis
• 4.5 Dimension
  • Dimension
• 4.6 Change of Basis
• 4.7 Row Space, Column Space, and Null Space
  • Row and Column Vector
  • Row Space, Column Space, Null Space
  • Dimension of Column and Row Space is Equal
• 4.8 Rank, Nullity, and the Fundamental Matrix Spaces
  • Rank and Nullity
  • Full Rank
• 4.9 Basic Matrix Transformations in R2 and R3
• 4.10 Properties of Matrix Transformations

Chapter 6: Inner Product Spaces

• 6.1 Inner Products
  • Inner Product
  • Real Inner Product Space
• 6.3 Gram-Schmidt Process; QR-Decomposition
  • Orthogonal and Orthonormal Sets
  • Linear Independence of Orthogonal Sets
  • Orthogonal Basis Expansion
  • Projection Theorem
  • The Gram-Schmidt Process

Collections

• Definitions