Similarity Mapping is Linear Operator
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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.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.3$