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.

$(1): \quad \forall x, y \in G: \map \phi {x +_G y} = \map \phi x +_H \map \phi y$

$(2): \quad \forall x \in G: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$

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.

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.

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 transformation.

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

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: $1$