# Definition:Linear Transformation

## Definition

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

Hence, let $R$ be a ring.

Let $M = \struct {G, +_G, \circ}_R$ and $N = \struct {H, +_H, \otimes}_R$ be $R$-modules.

Let $\phi: G \to H$ be a mapping.

Then $\phi$ is a **linear transformation** if and only if:

- $(1): \quad \forall x, y \in G: \map \phi {x +_G y} = \map \phi x +_H \map \phi y$
- $(2): \quad \forall x \in G: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$

### 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 denoted as

It is commonplace in the literature devoted to **linear transformations** for the argument not to be put in parenthesis:

That is, $A h$ would be used for $\map A h$, as long as the context makes this clear.

## 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**.

Some authors use the term **linear functional**, especially in the field of category theory.

## Also see

- Results about
**linear transformations**can be found here.

## Sources

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

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