Topics for Final

Algebraic structures

• Field

• Vector space

• Ring

• Algebra

• Group

• Module

Mappings

• Linear transformation of vector spaces (more on this below)

• Homomorphism of rings

• Homomorphism of algebras

• Homomorphism of groups

• Homomorphism of modules

Vector spaces

• Independence and span

• Basis and dimension

• Subspace

• Sum and intersection

• Direct sum; interpretation in terms of idempotents

• Coordinatization

• Change of basis

• Quotient space

Linear transformations (generalities)

• Matrix representation, similarity

• Kernel and image

• Rank/nullity theorem

• Induced map on dual spaces

• Quotient maps

Dual spaces

• Canonical pairing

• Dual basis

• Double dual

• Annihilators

Rings and algebras

• Polynomials (definition, degree,...)

• Ideals

• Lagrange interpolation

• Division algorithm

• Roots and linear factors

• Every ideal is principal in certain rings

• Greatest common divisors

• Unique factorization

• Factorization over algebraically closed fields

• Quotient rings

• Root adjunction

• Chinese remainder theorem

Determinants

• Multilinear alternating functions

• Existence and uniqueness of determinants

• Cofactor expansions

• Explicit formula with one term for each permutation

• Criterion for invertibility

• Cramer's rule

Universal mapping properties

• Vector space with basis

• Direct sum

• Quotient

• Polynomial algebra

• Tensor and wedge products

Multilinear algebra

• Definition and examples of tensor and wedge products
• Interpretation in terms of bases

Analysis of a single linear operator

• Characteristic and minimal polynomial; Cayley-Hamilton
• Eigenvalues, eigenvectors, diagonalizability, triangularizability
• Invariant subspaces, invariant direct sum decompositions
• Cyclic subspaces, T-annihilator
• Primary decomposition theorem
• Cyclic decomposition theorem
• Jordan normal form

Inner product spaces

• Cauchy-Schwarz
• Construction of orthogonal/orthonormal bases (Gram-Schmidt)
• "Fourier" expansions
• Orthogonal decomposition
• Isomorphism with dual
• Adjoint of a linear transformation
• Spectral theorem