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.

Depends on

Used by

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Sources

J. Lebl, Guide to Cultivating Complex Analysis, Section 1.1.1 (standard reference, not scraped)
E. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis II, Chapter 1, Section 1.1 (standard reference, not scraped)