Appendix B Definitions
1.1 Sets and functions
Definition 1.1.1 Sets
Definition 1.1.2 Set equality
Definition 1.1.3 Set inclusion (subsets)
Definition 1.1.5 Set-builder notation
Definition 1.1.8 Basic set operations
Definition 1.1.9 Common mathematical sets
Definition 1.1.10 Cartesian product
Definition 1.1.13 Functions
Definition 1.1.17 Function equality
Definition 1.1.18 Image of a set
Definition 1.1.19 Injective, surjective, bijective
Definition 1.1.21 Function composition
Definition 1.1.22 Identity and inverse functions
1.2 Logic
Definition 1.2.1 Logical operators
2.1 Systems of linear equations
Definition 2.1.1 Linear equations
Definition 2.1.3 Systems of linear equations
Definition 2.1.5 Solutions to linear systems
Definition 2.1.8 Elementary operations on linear systems
Definition 2.1.11 Equivalence of linear systems
2.2 Gaussian elimination
Definition 2.2.1 Augmented matrix
Definition 2.2.3 Row echelon form
Definition 2.2.5 Elementary row operations on matrices
Definition 2.2.8 Gaussian elimination
Definition 2.2.9 Gauss-Jordan elimination
2.3 Solving linear systems
Definition 2.3.1 Free and leading variables
Definition 2.3.4 Consistent and inconsistent systems
3.1 Matrices and their arithmetic
Definition 3.1.2 Matrix
Definition 3.1.5 Matrix equality
Definition 3.1.7 Square matrices, row vectors, column vectors, zero matrices
Definition 3.1.9 Matrix addition and subtraction
Definition 3.1.11 Scalar multiplication of matrices
Definition 3.1.13 Linear combination of matrices
Definition 3.1.17 Matrix multiplication
Definition 3.1.24 Matrix transposition
3.2 Algebra of matrices
Definition 3.2.2 Additive inverse of a matrix
Definition 3.2.3 Identity matrix
3.3 Invertible matrices
Definition 3.3.1 Invertible matrix
Definition 3.3.9 Matrix powers
3.4 The invertibility theorem
3.5 The determinant
Definition 3.5.1 Submatrix notation
Definition 3.5.3 The determinant
Definition 3.5.7 Minors and expansions along rows/columns
Definition 3.5.14 Adjoint matrix
4.1 Real vector spaces
Definition 4.1.1 Vector space
Definition 4.1.3 Vector space of \(m\times n\) matrices
Definition 4.1.4 Vector space of real \(n\)-tuples
Definition 4.1.6 Zero vector space
Definition 4.1.7 The vector space of infinite real sequences
Definition 4.1.8 Real-valued functions
Definition 4.1.10 Vector space of positive real numbers
Definition 4.1.11 Linear combination of vectors
4.2 Linear transformations
Definition 4.2.1 Linear transformations
Definition 4.2.3 Zero and identity transformation
Definition 4.2.8 Matrix transformations
Definition 4.2.12 Rotation in the plane
Definition 4.2.16 Reflection through a line
4.3 Subspaces
Definition 4.3.1 Subspace
Definition 4.3.12 Null space and image
Definition 4.3.19 Symmetric and skew-symmetric matrices
4.4 Span and linear independence
Definition 4.4.1 Span
Definition 4.4.5 Spanning set
Definition 4.4.8 Linear independence
4.5 Bases and dimension
Definition 4.5.1 Basis
Definition 4.5.9 Cardinality of a set
Definition 4.5.12 Dimension of a vector space
4.6 Rank-nullity theorem and fundamental spaces
Definition 4.6.1 Rank and nullity
Definition 4.6.5 Fundamental spaces
4.7 Isomorphisms
Definition 4.7.6 Isomorphism
5.1 Inner product spaces
Definition 5.1.1 Inner product
Definition 5.1.13 Norm (or length) of a vector
Definition 5.1.15 Distance between vectors
Definition 5.1.19 Angle between vectors
5.2 Orthogonal bases and orthogonal projection
Definition 5.2.1 Orthogonal
Definition 5.2.3 Orthogonal and orthonormal bases
Definition 5.2.8 Orthogonal matrices
Definition 5.2.9 Orthogonal complement
6.1 Coordinate vectors and isomorphisms
Definition 6.1.1 Ordered bases
Definition 6.1.3 Coordinate vectors
6.2 Matrix representations of linear transformations
6.3 Change of basis
Definition 6.3.1 Change of basis matrix
6.4 Eigenvectors and eigenvalues
6.5 Diagonalization