Category:Definitions/Linear Transformations

This category contains definitions related to Linear Transformations.
Related results can be found in Category:Linear Transformations.

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:

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

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