Topology/Morphisms

Topology Morphisms Vector Spaces Convexity In general, a morphism refers to a function mapping from one space to another that preserves structure. In terms of vector spaces the natural morphism is a linear map. Linear maps Definition of Linear Map A linear map is a function where are vector spaces over a field F. Such that for all 1. 2. The image of a linear map is a subspace of the domain. The kernel of a linear map is a subspace of the codomain. The Importance of Kernels and Images Definition of Rank The rank of a linear map is the dimension of the image of the map . It also can be found using row reduction on the corresponding matrix. Definition of Nullity The nullity of a linear map, or matrix, is the dimension of the kernel of the map . The Rank-Nullity Theorem For any linear map Topology Morphisms Vector Spaces Convexity