Section B.3 Lemmas, Propositions, and Theorems
This section provides a comprehensive list of all mathematical theorems throughout the text.
1 Welcome!
Axiom 1.2.3 The Division Algorithm
Theorem 1.2.11 Greatest common divisors with remainders
Theorem 1.2.13 Greatest common divisor is a linear combination
2 Permutations
Theorem 2.0.1 Multiplicative Principle
Theorem 2.1.8 \(\sym{A}\) is closed under composition
Theorem 2.1.10 Composition in \(\sym{A}\) is associative
Theorem 2.1.17 Unique inverses
Theorem 2.1.23 Cayley’s Theorem
Lemma 2.1.28 Disjoint cyclic permutations commute
Theorem 2.1.29 Disjoint cycle decomposition
4 Linear Algebra
Theorem 4.1.4 \(\CV{n}\) is a vector space
Theorem 4.1.10 Every complex vector space is a \(\CV{n}\)
Theorem 4.1.14 Hermitian inner product and magnitude
Theorem 4.3.2 Matrices form a vector space over \(\Comps\)
Theorem 4.3.6 Components of a matrix-vector product
Theorem 4.3.7 Matrix-vector products are linear transformations
Theorem 4.3.8 Matrix representation of a transformation
Theorem 4.3.10 Entries of a matrix-matrix product
Theorem 4.3.11 Matrix-matrix products do composition
Theorem 4.4.4 Certain matrix products always exist
5 Matrix Applications
Theorem 5.1.7 Elementary matrices
Theorem 5.1.9 Row-equivalent augmented matrices have the same solutions
Theorem 5.1.12 Existence and Uniqueness of RREF
Theorem 5.1.15 Matrix inverse uniqueness
Theorem 5.1.16 Elementary matrices are invertible
Theorem 5.1.17 GJE produces matrix inverses
Theorem 5.3.4 Solution via forward substitution
Theorem 5.3.5 Solution via back substitution
6 Graph Theory Algorithms
Theorem 6.4.3 Max-flow Min-cut Theorem
