In mathematics, and more specifically in linear algebra, a linear map (or linear mapping) 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.

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.

Let ${\displaystyle V}$ and ${\displaystyle W}$ be vector spaces over the same field ${\displaystyle K}$, such as the real or complex numbers. A function ${\displaystyle f:V\to W}$ is said to be a linear map if for any two vectors ${\textstyle \mathbf {u} ,\mathbf {v} \in V}$ and any scalar ${\displaystyle c\in K}$ the following two conditions are satisfied:

Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.

By the associativity of the addition operation denoted as +, for any vectors ${\textstyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}\in V}$ and scalars ${\textstyle c_{1},\ldots ,c_{n}\in K,}$ the following equality holds: $${\displaystyle f(c_{1}\mathbf {u} _{1}+\cdots +c_{n}\mathbf {u} _{n})=c_{1}f(\mathbf {u} _{1})+\cdots +c_{n}f(\mathbf {u} _{n}).}$$ Thus a linear map is one which preserves linear combinations.

Denoting the zero elements of the vector spaces ${\displaystyle V}$ and ${\displaystyle W}$ by ${\textstyle \mathbf {0} _{V}}$ and ${\textstyle \mathbf {0} _{W}}$ respectively, it follows that ${\textstyle f(\mathbf {0} _{V})=\mathbf {0} _{W}.}$ Let ${\displaystyle c=0}$ and ${\textstyle \mathbf {v} \in V}$ in the equation for homogeneity of degree 1: $${\displaystyle f(\mathbf {0} _{V})=f(0\mathbf {v} )=0f(\mathbf {v} )=\mathbf {0} _{W}.}$$

A linear map ${\displaystyle V\to K}$ with ${\displaystyle K}$ viewed as a one-dimensional vector space over itself is called a linear functional.

These statements generalize to any left-module ${\textstyle {}_{R}M}$ over a ring ${\displaystyle R}$ without modification, and to any right-module upon reversing of the scalar multiplication.