Appendix C Definitions
0.1 Sets
Definition 0.1.1 Sets
Definition 0.1.2 Set equality
Definition 0.1.3 Set inclusion (subsets)
Definition 0.1.5 Set-builder notation
Definition 0.1.8 Union, intersection, difference, and complement
Definition 0.1.9 Common mathematical sets
0.2 Functions
Definition 0.2.1 Functions
Definition 0.2.5 Function equality
Definition 0.2.6 Image of a set
Definition 0.2.7 Injective, surjective, bijective
Definition 0.2.9 Function composition
Definition 0.2.10 Identity and inverse functions
0.3 Tuples and Cartesian products
Definition 0.3.1 \(n\)-tuple
Definition 0.3.4 Cartesian product (finite)
Definition 0.3.5 I-tuple
Definition 0.3.6 Cartesian product (arbitrary)
0.4 Logic
Definition 0.4.1 Logical operators
0.6 Complex numbers
Definition 0.6.1 Complex numbers
Definition 0.6.4 Complex addition and multiplication
Definition 0.6.9 Absolute value and complex conjugation
0.7 Polynomials
Definition 0.7.1 Polynomials
Definition 0.7.5 Degree of a polynomial
1.1 Systems of linear equations
Definition 1.1.1 Linear equations
Definition 1.1.3 Systems of linear equations
Definition 1.1.5 Solutions to linear systems
Definition 1.1.11 Elementary operations on linear systems
Definition 1.1.12 Row equivalent linear systems
1.2 Gaussian elimination
Definition 1.2.1 Augmented matrix
Definition 1.2.3 Row echelon form
Definition 1.2.5 Elementary row operations on matrices
Definition 1.2.8 Gaussian elimination
Definition 1.2.9 Gauss-Jordan elimination
1.3 Solving linear systems
Definition 1.3.1 Free and leading variables
Definition 1.3.4 Consistent and inconsistent systems
2.1 Matrix arithmetic
Definition 2.1.2 Matrix
Definition 2.1.5 Matrix equality
Definition 2.1.7 Square matrices, row vectors, column vectors, zero matrices
Definition 2.1.9 Matrix addition and subtraction
Definition 2.1.11 Scalar multiplication of matrices
Definition 2.1.13 Linear combination of matrices
Definition 2.1.17 Matrix multiplication
Definition 2.1.21 Dot product
Definition 2.1.29 Matrix transposition
2.2 Matrix algebra
Definition 2.2.2 Additive inverse of a matrix
Definition 2.2.3 Identity matrix
2.3 Invertible matrices
Definition 2.3.1 Invertible matrix
Definition 2.3.9 Matrix powers
2.4 The invertibility theorem
Definition 2.4.1 Elementary matrices
Definition 2.4.7 Diagonal and triangular matrices
2.5 The determinant
Definition 2.5.1 Submatrix notation
Definition 2.5.3 The determinant
Definition 2.5.7 Minors and expansions along rows/columns
Definition 2.5.15 Adjoint matrix
3.1 Real vector spaces
Definition 3.1.1 Vector space
Definition 3.1.3 Vector space of \(m\times n\) matrices
Definition 3.1.4 Vector space of real \(n\)-tuples
Definition 3.1.6 Zero vector space
Definition 3.1.7 The vector space of infinite real sequences
Definition 3.1.8 Real-valued functions
Definition 3.1.10 Vector space of positive real numbers
Definition 3.1.11 Linear combination of vectors
3.2 Linear transformations
Definition 3.2.1 Linear transformations
Definition 3.2.3 Zero and identity transformation
Definition 3.2.8 Matrix transformations
Definition 3.2.12 Rotation in the plane
Definition 3.2.16 Reflection through a line
3.3 Subspaces
Definition 3.3.1 Subspace
Definition 3.3.15 Trace of a matrix
Definition 3.3.16 Trace-zero, symmetric, and skew-symmetric
3.4 Null space and image
Definition 3.4.1 Null space and image
Definition 3.4.5 Null space of a matrix
3.5 Span and linear independence
Definition 3.5.1 Span
Definition 3.5.6 Spanning set
Definition 3.5.9 Linear independence (for finite subsets)
Definition 3.5.14 Linear independence (for arbitrary subsets)
3.6 Bases
Definition 3.6.1 Basis
Definition 3.6.17 Standard matrix of linear \(T\colon \R^n\rightarrow \R^m\)
3.7 Dimension
Definition 3.7.1 Cardinality of a set
Definition 3.7.4 Dimension of a vector space
3.8 Rank-nullity theorem and fundamental spaces
Definition 3.8.1 Rank and nullity
Definition 3.8.5 Fundamental spaces
3.9 Isomorphisms
Definition 3.9.1 Isomorphism
4.1 Inner product spaces
Definition 4.1.1 Inner product
Definition 4.1.4 (Weighted) Euclidean space
Definition 4.1.14 Norm (or length) of a vector
Definition 4.1.20 Distance between vectors
Definition 4.1.25 Angle between vectors
4.2 Orthogonal bases
Definition 4.2.1 Orthogonality
Definition 4.2.5 Orthogonal and orthonormal bases
4.3 Orthogonal projection
Definition 4.3.1 Orthogonal complement
5.1 Coordinate vectors
Definition 5.1.1 Ordered bases
Definition 5.1.4 Coordinate vectors
5.2 Matrix representations of linear transformations
Definition 5.2.1 Matrix representations of linear transformations
5.3 Change of basis
Definition 5.3.1 Change of basis matrix
Definition 5.3.12 Orthogonal matrices
5.4 Eigenvectors and eigenvalues
Definition 5.4.3 Eigenvectors and eigenvalues
Definition 5.4.15 Eigenspaces
Definition 5.4.18 Characteristic polynomial of a matrix
5.5 Diagonalization
Definition 5.5.1 Diagonalizable
Definition 5.5.3 Eigenbasis
Definition 5.5.27 Characteristic polynomial of a transformation
Definition 5.5.29 Algebraic/geometric multiplicity
5.6 The spectral theorem
Definition 5.6.1 Self-adjoint operators
Definition 5.6.9 Orthogonally diagonalizable
