# Similarity Mapping is Linear Operator

## Theorem

Let $G$ be a vector space over a field $\struct {K, + \times}$.

Let $\beta \in K$.

Then the similarity $s_\beta: G \to G$ defined as:

$\forall \mathbf x \in G: \map {s_\beta} {\mathbf x} = \beta \mathbf x$

is a linear operator on $G$.

## Proof

To prove that $s_\beta$ is a linear operator it is sufficient to demonstrate that:

$(1): \quad \forall \mathbf x, \mathbf y \in G: \map {s_\beta} {\mathbf x + \mathbf y} = \map {s_\beta} {\mathbf x} + \map {s_\beta} {\mathbf y}$
$(2): \quad \forall \mathbf x \in G: \forall \lambda \in K: \map {s_\beta} {\lambda \mathbf x} = \lambda \map {s_\beta} {\mathbf x}$

Indeed:

 $\ds \forall \mathbf x, \mathbf y \in G: \,$ $\ds \map {s_\beta} {\mathbf x + \mathbf y}$ $=$ $\ds \beta \paren {\mathbf x + \mathbf y}$ Definition of $s_\beta$ $\ds$ $=$ $\ds \beta \, \mathbf x + \beta \, \mathbf y$ Vector Space Axiom $\text V 6$: Distributivity over Vector Addition $\ds$ $=$ $\ds \map {s_\beta} {\mathbf x} + \map {s_\beta} {\mathbf y}$ Definition of $s_\beta$

and:

 $\ds \forall \mathbf x \in G: \forall \lambda \in K: \,$ $\ds \map {s_\beta} {\lambda \mathbf x}$ $=$ $\ds \beta \paren {\lambda \mathbf x}$ Definition of $s_\beta$ $\ds$ $=$ $\ds \beta \lambda \paren {\mathbf x}$ Vector Space Axiom $\text V 7$: Associativity with Scalar Multiplication $\ds$ $=$ $\ds \lambda \beta \paren {\mathbf x}$ Field Axiom $\text M2$: Commutativity of Product $\ds$ $=$ $\ds \lambda \map {s_\beta} {\mathbf x}$ Definition of $s_\beta$

Hence the result.

$\blacksquare$