# Definition:Linear Transformation/Vector Space

## Definition

Let $V, W$ be vector spaces over a field (or, more generally, division ring) $K$.

A mapping $A: V \to W$ is a linear transformation if and only if:

$\forall v_1, v_2 \in V, \lambda \in K: \map A {\lambda v_1 + v_2} = \lambda \map A {v_1} + \map A {v_2}$

That is, a homomorphism from one vector space to another.

### Linear Operator on Vector Space

A linear operator on a vector space is a linear transformation from a vector space into itself.

## Also known as

A linear transformation is also known as a linear mapping.