Linear map

In mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication. A standard example of a linear map is an ${\displaystyle m\times n}$ matrix, which takes vectors in ${\displaystyle n}$-dimensions into vectors in ${\displaystyle m}$-dimensions in a way that is compatible with addition of vectors, and multiplication of vectors by scalars. When the two vector spaces are the same, a linear map is also called a linear transformation or linear endomorphism.

A linear map is a homomorphism of vector spaces. Thus, a linear map ${\displaystyle T:V\to W}$ satisfies ⁠${\displaystyle T(ax+by)=aTx+bTy}$⁠, where ${\displaystyle a}$ and ${\displaystyle b}$ are scalars, and ${\displaystyle x}$ and ${\displaystyle y}$ are vectors (elements of the vector space ⁠${\displaystyle V}$⁠). A linear mapping always maps the origin of ${\displaystyle V}$ to the origin of ⁠${\displaystyle W}$⁠, and linear subspaces of ${\displaystyle V}$ onto linear subspaces in ${\displaystyle W}$ (possibly of a lower dimension); for example, it maps a plane through the origin in ${\displaystyle V}$ to either a plane through the origin in ⁠${\displaystyle W}$⁠, a line through the origin in ⁠${\displaystyle W}$⁠, or just the origin in ⁠${\displaystyle W}$⁠. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.