Website owner:  James Miller

Linear algebra

Linear space (abstract vector space)

Linear spaces --- examples

Properties that follow from the axiomatic definitions of different spaces

Linear algebra

The concepts of linear algebra represent the end result of a series of generalizations and abstractions

Abstract vector spaces

Motivation for extending the concept of a vector to include objects of an arbitrary nature. Generalization of the concept of a vector space

Vector space, subspace, basis, dimension, linear independence

Normed linear spaces

Space of polynomials

Elementary bases for various linear spaces

Coordinates of a vector in an abstract vector space. Coordinate vector.

Functions, mappings, maps, transformations, operators

Linear mapping (or linear transformation)

Linear transformations, examples

Linear transformations, linear mappings, linear operators are linear vector functions assigning objects to objects

Matrix representation of a linear transformation

Linear transformation, linear mapping. Operations, sum, product. Algebra of linear operators. Invertible operators.

Hom(v,w). Vector space of all mxn matrices

Concept of an operator

Eigenvectors and eigenvalues

Linear mappings. Hom(V, W). Image, kernel, rank, nullity. Singular and nonsingular mappings.

Synonyms

Linear operators

Isomorphisms and homomorphisms effected by linear transformations from one abstract vector space into another

In linear transformations between abstract n-dimensional vector spaces the operators are always matrices

Vector space Hom(V,W)

Inner product and inner product space

Examples of inner products defined on various abstract spaces

Orthogonality

Projection of a vector on a subspace

Sums and direct sums of vector subspaces

Algebra over a field

Linear functional. Matrix representation. Dual space, conjugate space, adjoint space. Basis for dual space. Annihilator. Transpose of a linear mapping.

Importance of concrete models in understanding many abstract concepts in mathematics