Definition:Linear Transformation

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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 an arbitrary linear transformation.

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

Also see

  • Results about linear transformations can be found here.