Contents
Preface
Systems of linear equations
Geometric view of systems of equations
Algebraic view of systems of equations
Elementary operations
Gaussian elimination
Gauss-Jordan elimination
Homogeneous systems
Uniqueness of the reduced echelon form
Fields
Application: Balancing chemical reactions
Application: Dimensionless variables
Application: Resistor networks
Vectors in Rn
Points and vectors
Addition
Scalar multiplication
Linear combinations
Length of a vector
The dot product
Definition and properties
The Cauchy-Schwarz and triangle inequalities
The geometric significance of the dot product
Orthogonal vectors
Projections
The cross product
Right-handed systems of vectors
Geometric description of the cross product
Algebraic definition of the cross product
The box product
Lines and planes in Rn
Lines
Planes
Matrices
Definition and equality
Addition
Scalar multiplication
Matrix multiplication
Multiplying a matrix and a vector
Matrix multiplication
Properties of matrix multiplication
Matrix inverses
Definition and uniqueness
Computing inverses
Using the inverse to solve a system of equations
Properties of the inverse
Right and left inverses
Elementary matrices
Elementary matrices and row operations
Inverses of elementary matrices
Elementary matrices and reduced echelon forms
Writing an invertible matrix as a product of elementary matrices
More properties of inverses
The transpose
Matrix arithmetic modulo p
Application: Cryptography
Spans, linear independence, and bases in Rn
Spans
Linear independence
Redundant vectors and linear independence
The casting-out algorithm
Alternative characterization of linear independence
Properties of linear independence
Linear independence and linear combinations
Removing redundant vectors
Subspaces of Rn
Basis and dimension
Definition of basis
Examples of bases
Bases and coordinate systems
Dimension
More properties of bases and dimension
Column space, row space, and null space of a matrix
Linear transformations in Rn
Linear transformations
The matrix of a linear transformation
Geometric interpretation of linear transformations
Properties of linear transformations
Application: Perspective rendering
Determinants
Determinants of 2x2- and 3x3-matrices
Minors and cofactors
The determinant of a triangular matrix
Determinants and row operations
Properties of determinants
Application: A formula for the inverse of a matrix
Application: Cramer's rule
Eigenvalues, eigenvectors, and diagonalization
Eigenvectors and eigenvalues
Finding eigenvalues
Geometric interpretation of eigenvectors
Diagonalization
Application: Matrix powers
Application: Solving recurrences
Application: Systems of linear differential equations
Differential equations
Systems of linear differential equations
Example: coupled train cars
Application: The matrix exponential
Properties of eigenvectors and eigenvalues
The Cayley-Hamilton Theorem
Complex eigenvalues and eigenvectors
Vector spaces
Definition of vector spaces
Linear combinations, span, and linear independence
Subspaces
Basis and dimension
Application: Error correcting codes
Linear transformation of vector spaces
Definition and examples
The algebra of linear transformations
Linear transformations defined on a basis
The matrix of a linear transformation
Inner product spaces
Real inner product spaces
Orthogonality
The Gram-Schmidt orthogonalization procedure
Orthogonal projections and Fourier series
Application: Least squares approximations and curve fitting
Orthogonal functions and orthogonal matrices
Diagonalization of symmetric matrices
Positive semidefinite and positive definite matrices
Application: Simplification of quadratic forms
Complex inner product spaces
Unitary and hermitian matrices
Application: Principal component analysis
Complex numbers
The complex numbers
Geometric interpretation
The fundamental theorem of algebra
Answers to selected exercises
Index