Condition for Linear Transformation

Theorem

Let $G$ be a unitary $R$-module.

Let $H$ be an $R$-module.

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

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

$\forall x, y \in G: \forall \lambda, \mu \in R: \map \phi {\lambda x + \mu y} = \lambda \map \phi x + \mu \map \phi y$

Proof

Sufficient Condition

\(\ds \forall x, y \in G: \forall \lambda, \mu \in R: \, \)

\(\ds \map \phi {\lambda x + \mu y}\)

\(=\)

\(\ds \map \phi {\lambda x} + \map \phi {\mu y}\)

Definition of Linear Transformation: condition $(1)$

\(\ds \)

\(=\)

\(\ds \lambda \map \phi x + \mu \map \phi y\)

Definition of Linear Transformation: condition $(2)$

$\Box$

Necessary Condition

Let $\phi$ be such that the condition is satisfied.

Let $\lambda = \mu = 1_R$.

Then:

$\map \phi {x + y} = \map \phi x + \map \phi y$

Now let $\mu = 0_R$.

Then:

$\map \phi {\lambda x} = \lambda \map \phi x$

Thus the conditions are fulfilled for $\phi$ to be a homomorphism, that is, a linear transformation.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations