Rescaling is Linear Transformation

Theorem

Let $\struct {R, +, \cdot}$ be a commutative ring.

Let $\struct {V, +, \circ}_R$ be an $R$-module.

Then for any $r \in R$, the rescaling:

$m_r: V \to V, v \mapsto r \circ v$

is a linear transformation.

Proof

Let $v \in V$ and $s \in R$.

Then:

\(\ds \map {m_r} {s \circ v}\)

\(=\)

\(\ds r \circ \paren {s \circ v}\)

Definition of Rescaling

\(\ds \)

\(=\)

\(\ds \paren {r \cdot s} \circ v\)

$V$ is an $R$-module

\(\ds \)

\(=\)

\(\ds \paren {s \cdot r} \circ v\)

$R$ is a commutative ring

\(\ds \)

\(=\)

\(\ds s \circ \paren {r \circ v}\)

$V$ is an $R$-module

\(\ds \)

\(=\)

\(\ds s \circ \map {m_r} v\)

Definition of Rescaling

Next, for $v, w \in V$:

\(\ds m_r \paren {v + w}\)

\(=\)

\(\ds r \circ \paren {v + w}\)

Definition of Rescaling

\(\ds \)

\(=\)

\(\ds r \circ v + r \circ w\)

$V$ is an $R$-module

\(\ds \)

\(=\)

\(\ds \map {m_r} v + \map {m_r} w\)

Definition of Rescaling

It follows that $m_r$ is a linear transformation.

$\blacksquare$