Coding theory and cryptography
The algebra that codes and lattices are built from: changing the field, rank over a finite field, irreducible polynomials, the Chinese remainder theorem, and the Smith normal form.
It aims at 1.9 Changing the Field, 3.5 Row Space, Column Space, Null Space and Rank, 6.3 Irreducible Polynomials and Unique Factorization, 6.6 Interpolation and the Chinese Remainder Theorem, 10.7 Polynomial Matrices and the Smith Normal Form.
65 sections to read, of 236. That is everything the proofs of those sections rest on, and nothing else.
70 if you also work the exercises, since a written solution is the proof of its exercise: 5 more sections.
What to read
In reading order. Bold marks a section the path is aimed at; + marks one that only the exercises need. The note beside a section says when the whole path hangs on a single proof citing it.
Preliminaries10 of 10
Vector Spaces9 of 9
Linear Maps5 of 5
Linear Systems and Matrices7 of 7
Matrices of Maps4 of 5
Duality1 of 6
Polynomials7 of 7
Determinants8 of 10
- 7.2 Multilinear Alternating Forms
- 7.3 Permutations and Sign
- 7.4 Existence and Uniqueness of the Determinant
- 7.5 Row Operations and Multiplicativity
- 7.6 Cofactor Expansion, the Adjugate and Cramer's Rule
- 7.7 Minors, Rank and Cauchy–Binet2 proofs
- 7.8 Special Determinants
- 7.10 The Characteristic Polynomial
Block Matrices4 of 5
Eigenvalues and Diagonalization9 of 11
Canonical Forms6 of 10
What you can skip
171 of the book's 236 sections are not on this path, and skipping them costs you no proof you need.
Not needed at all: Inner Product Spaces, Spectral Theory, Positive Matrices and the Singular Value Decomposition, Bilinear and Quadratic Forms, Tensors and Exterior Algebra, Norms and Matrix Analysis, Variational Principles and Interlacing, Convexity, Non-negative Matrices, Perturbation Theory, Matrix Inequalities, Affine and Projective Geometry, Algebras and Representations, Computation and Applications.
Computed from the book's proofs when the book was built, and checked: every result a proof on this path uses is proved by another section on this path. The other paths are listed on the reading paths page.