TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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The complex coordinate map preserves real scalar multiplication

Statement

Let $Φ : C \to R^{2}$ be the coordinate bijection $Φ (a + b i) = (a, b)$ . Here a map is real-linear when it preserves addition and multiplication by real scalars embedded in $C$ via The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ . Then $Φ$ is real-linear: for all $z, w \in C$ and $r \in R$ ,

$Φ (z + w) = Φ (z) + Φ (w), Φ (r z) = r Φ (z) .$

Facts & Assumptions

Given: The complex field and the coordinate map $Φ$ .

[F1]

$C$ is the real coordinate plane, with coordinate arithmetic: Each complex number has a unique form $a + b i$ ; $Φ$ is bijective and preserves addition, and $Φ ((a + b i) (u + v i)) = (a u - b v, a v + b u)$ .

[F2]

The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ : The canonical real embedding is the constant-class map; the imaginary unit is the class of $x$ , so an embedded real $r$ has coordinate form $r + 0 i$ .

Proof

1.1givenF1F2algebra

Write $z = a + b i$ . The embedded scalar has coordinates $(r, 0)$ by [F2], so the coordinate multiplication formula in [F1] gives $Φ (r z) = (r a, r b) = r (a, b) = r Φ (z)$ for every $r$ , including $r = 0$ .

2.1F1step 1.1∎

Additivity and bijectivity are [F1]; step 1.1 supplies real homogeneity. These are exactly the stated real-linearity conditions, so $Φ$ is a real-linear bijection.

The complex coordinate map preserves real scalar multiplication

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources