Similarity Mapping is Linear Operator

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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 $G$.


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

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


\(\ds \forall \mathbf x, \mathbf y \in G: \, \) \(\ds \map {s_\beta} {\mathbf x + \mathbf y}\) \(=\) \(\ds \beta \paren {\mathbf x + \mathbf y}\) 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 $s_\beta$


\(\ds \forall \mathbf x \in G: \forall \lambda \in K: \, \) \(\ds \map {s_\beta} {\lambda \mathbf x}\) \(=\) \(\ds \beta \paren {\lambda \mathbf x}\) 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.