# Definition:Linear Transformation/Vector Space

< Definition:Linear Transformation(Redirected from Definition:Linear Transformation on Vector Space)

Jump to navigation
Jump to search
## 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**.

## Sources

- 2013: Francis Clarke:
*Functional Analysis, Calculus of Variations and Optimal Control*... (previous) ... (next): $1.2$: Linear mappings

- For a video presentation of the contents of this page, visit the Khan Academy.