Similarity Mapping on Plane with Scale Factor Minus 1
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Theorem
Let $s_{-1}: \R^2 \to \R^2$ be a similarity mapping on $\R^2$ whose scale factor is $-1$.
Then $s_{-1}$ is the same as the rotation $r_\pi$ of the plane about the origin one half turn.
Proof
Let $P = \tuple {x, y} \in \R^2$ be an aribtrary point in the plane.
Then:
\(\ds \map {r_\pi} P\) | \(=\) | \(\ds \tuple {\paren {\cos \pi - \sin \pi} x, \paren {\sin \pi + \cos \pi} y}\) | Rotation of Plane about Origin is Linear Operator | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {\paren {\paren {-1} - 0} x, \paren {0 + \paren {-1} } y}\) | Cosine of Straight Angle, Sine of Straight Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {-x, -y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1} \tuple {x, y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {s_{-1} } P\) | Definition of $s_{-1}$ |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.3$