# Definition:Linear Transformation

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## Definition

A **linear transformation** is a homomorphism from one module to another.

### Definition in a Vector Space

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

A **linear operator** is a linear transformation from a module into itself.

## Also known as

The term **linear mapping** can sometimes be found, which means the same thing as **linear transformation**.

Some sources use the term **module homomorphism**.

Some authors, specifically in the field of functional analysis, use the term **linear operator** (or even just **operator**) for arbitrary **linear transformations**.

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $1$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 28$

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