Similarity Mapping on Plane with Scale Factor Minus 1

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 $28.3$