Algebra

The study of mathematical structures and their operations.

Linear Algebra

• Vector — Directed magnitude
• Vector Addition — Parallelogram rule
• Scalar Multiplication — Scaling vectors
• Dot Product — Inner product on R^n
• Cross Product — Orthogonal product in R^3
• Vector Space — Axiomatic structure
• Subspace — Subset that is itself a vector space
• Linear Combination — Sum of scaled vectors
• Span — All possible linear combinations
• Linear Independence — No redundant vectors
• Basis — Minimal spanning set
• Dimension — Size of any basis
• Linear Map — Structure-preserving transformation
• Kernel — Elements mapping to zero
• Image — Range of a linear map
• Rank-Nullity Theorem — dim(V) = rank + nullity
• Matrix — Rectangular array
• Matrix Addition — Entrywise sum
• Matrix Multiplication — Row by column
• Transpose — Rows become columns
• Inverse Matrix — Matrix that undoes another
• Determinant — Scaling factor of volumes
• Minor — Submatrix determinant
• Cofactor — Signed minor
• Cofactor Expansion — Laplace expansion for determinants
• Eigenvalue — Scalar λ where Av = λv
• Eigenvector — Nonzero vector in the eigenspace
• Characteristic Polynomial — det(A - λI)
• Diagonalization — A = PDP^{-1}
• Inner Product Space — General inner product
• Orthogonality — Perpendicular vectors
• Orthogonal Basis — Basis of orthogonal vectors
• Orthonormal Basis — Basis of unit orthogonal vectors
• Gram-Schmidt Process — Constructing orthonormal bases

Abstract Algebra

• Group — Set with associative binary operation and inverses
• Subgroup — Subset that forms a group
• Coset — aH for subgroup H
• Normal Subgroup — aH = Ha for all a
• Quotient Group — G/N
• Group Homomorphism — Structure-preserving map
• Kernel (Homomorphism) — Preimage of identity
• Isomorphism Theorem — First isomorphism theorem
• Ring — Two operations, distributive
• Ideal — Absorbing subset of a ring
• Quotient Ring — R/I
• Field — Commutative division ring
• Polynomial — Formal expression in one variable
• Polynomial Ring — R[x]
• Root — Where polynomial evaluates to zero
• Factorization — Breaking into irreducibles
• Irreducible — Cannot factor further
• Euclidean Domain — Division algorithm works