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3.2. Algebra and Number Theory (ANT)

This area covers the algebraic structures essential for computing: linear algebra for machine learning and graphics, abstract algebra for cryptography and coding theory, and the number-theoretic foundations of modern security protocols.

Knowledge Area (KA)	CS Core	KA Core
3.2.1 Matrices and Systems of Linear Equations	1	1
3.2.2 Vector Spaces and Linear Transformations	1	1
3.2.3 Eigenvalues, Inner Product Spaces, and Quadratic Forms	1	1
3.2.4 Group Theory	1	1
3.2.5 Rings, Fields, and Galois Theory	1	1
3.2.6 Cryptography and Coding Theory	1	1

3.2.1. ANT/Matrices and Systems of Linear Equations  (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top

Matrix algebra, systems of linear equations and their solution methods, determinants, and matrix decompositions, with applications to data science and computer graphics.
Topics:
Core

• Matrix algebra: addition, scalar multiplication, matrix multiplication, transpose, types of matrices, and the inverse of a matrix
• Solving systems of linear equations via Gaussian elimination and Gauss-Jordan; row echelon form, rank, and consistency of systems
• Determinants: definition, properties, Cramer's rule, and geometric interpretation
• Matrix decompositions: LU, QR, and singular value decomposition (SVD)

Learning Outcomes:
Core:

1. Identify types of matrices and perform basic matrix algebra operations, including computing the inverse of a matrix [Familiarity]
2. Solve systems of linear equations using Gaussian elimination or Gauss-Jordan, determining the rank and consistency of the system [Usage]
3. Apply Cramer's rule and the LU, QR, and SVD decompositions to solve linear systems and factor matrices [Assessment]

3.2.2. ANT/Vector Spaces and Linear Transformations  (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top

Vector spaces, subspaces, bases, and linear maps between vector spaces, with applications to machine learning.
Topics:
Core

• Vector spaces and subspaces; bases, dimension, and linear independence
• Linear maps, kernel and image, rank-nullity theorem, and matrix representations

Learning Outcomes:
Core:

1. Identify whether a set with given operations forms a vector space and determine a basis [Familiarity]
2. Determine the kernel, image, and matrix representation of a linear map, verifying the rank-nullity theorem [Usage]
3. Apply linear transformations to model and solve problems involving change of basis and composition of maps [Assessment]

3.2.3. ANT/Eigenvalues, Inner Product Spaces, and Quadratic Forms  (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top

Eigenvalues, diagonalization, inner product spaces, and quadratic forms, with applications to machine learning and data science.
Topics:
Core

• Eigenvalues and eigenvectors; diagonalization and the spectral theorem
• Inner product spaces, orthogonality, Gram-Schmidt process, and least squares
• Quadratic forms and definiteness: classification as positive, negative, or indefinite via eigenvalues

Learning Outcomes:
Core:

1. Identify the eigenvalues and eigenvectors of a matrix and determine whether it is diagonalizable [Familiarity]
2. Compute the diagonalization of matrices via the spectral theorem and apply the Gram-Schmidt process to obtain orthogonal bases [Usage]
3. Apply least squares methods and the classification of quadratic forms via eigenvalues to solve approximation and optimization problems [Assessment]

3.2.4. ANT/Group Theory  (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top

Groups, subgroups, homomorphisms, quotient groups, and the classification of finite groups, with applications to symmetry in computing and cryptography.
Topics:
Core

• Groups, subgroups, cyclic groups, and order of elements
• Cosets, Lagrange's theorem, and normal subgroups
• Group homomorphisms, isomorphisms, and the isomorphism theorems
• Group actions, orbits, stabilizers, and Burnside's lemma

Learning Outcomes:
Core:

1. State Lagrange's theorem and explain the relationship between subgroups and cosets [Familiarity]
2. Apply the isomorphism theorems to analyze homomorphic images and quotient groups [Usage]
3. Analyze symmetry groups arising in computing contexts such as permutation groups over data structures [Assessment]

3.2.5. ANT/Rings, Fields, and Galois Theory  (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top

Rings, ideals, polynomial rings, field extensions, and finite fields foundational for coding theory and cryptography.
Topics:
Core

• Rings, ideals, quotient rings, and the ring isomorphism theorems
• Polynomial rings: irreducibility, factorization, and unique factorization domains
• Field extensions: algebraic and transcendental elements, degree, and splitting fields
• Finite fields: structure, existence, and applications in coding theory

Learning Outcomes:
Core:

1. Classify rings as integral domains, PIDs, or UFDs and justify the classification [Familiarity]
2. Construct and analyze finite fields used in AES and other cryptographic algorithms [Usage]
3. Apply field extension theory to analyze error-correcting codes over finite fields [Assessment]

3.2.6. ANT/Cryptography and Coding Theory  (CS Core: 1 hr, KA Core: 1 hr) ↑ Back to top

Mathematical foundations of modern cryptography and error-correcting codes, including RSA, elliptic curves, and linear codes.
Topics:
Core

• Modular arithmetic, the RSA cryptosystem, and discrete logarithm problem
• Elliptic curves over finite fields and elliptic curve cryptography (ECC)
• Linear codes: generator and parity-check matrices, Hamming distance, and error correction
• Cyclic codes: polynomial representation, BCH codes, and Reed-Solomon codes

Non Core

• Post-quantum cryptography: lattice-based schemes and the Learning With Errors (LWE) problem

Learning Outcomes:
Core:

1. Explain the mathematical hardness assumptions underlying RSA and elliptic curve cryptography [Familiarity]
2. Construct a linear code, compute its parameters, and determine its error-correction capability [Usage]

NonCore:

1. Analyze the security assumptions of lattice-based cryptographic schemes in the post-quantum context [Assessment]