kernel

Kernel and image of a linear map are the two fundamental subspaces associated with a linear transformation. The kernel contains all vectors mapped to the zero vector, while the image consists of all vectors that can be obtained as outputs of the transformation.

Groups are algebraic structures consisting of a set equipped with a binary operation satisfying closure, associativity, identity, and invertibility.

Homomorphisms and isomorphisms are functions that preserve algebraic structure, allowing groups, rings, fields, and modules to be compared.

A field is an algebraic structure in which addition and multiplication are defined and invertible, except for division by zero. It generalizes number systems such as rational, real, and complex numbers, and forms a foundational concept in algebra.

A vector space is an algebraic structure consisting of a set of vectors together with a field of scalars, where vector addition and scalar multiplication satisfy specific axioms. It provides the foundation for linear algebra, including concepts such as subspaces, basis, dimension, and linear transformations.