Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers

Since the real numbers together with addition and multiplication form a field (\(\mathbb R,+,\cdot)\) and the complex numbers form a non-empty set, we can define the maps

\[ \cases{ \mathbb C\times \mathbb C\mapsto \mathbb C:x,y\mapsto x + y:=(a+c,b+d)  & \text{(the vector addition)}\\ \mathbb R\times \mathbb C\mapsto \mathbb C:\alpha,x\mapsto \alpha \cdot x:=(\alpha \cdot a,\alpha \cdot b) & \text{(the scalar multiplication)} }\]

for any real number \(\alpha\in\mathbb R\) and any complex numbers \(x,y\in\mathbb C\), identified by some ordered pairs of real numbers \(x:=(a,b)\), \(y:=(c,d)\), \(a,b,c,d\in\mathbb R\). Therefore, we can consider the complex numbers \(\mathbb C\) as a vector space over the field \(\mathbb R\).

Proofs: 1

Lemmas: 1 2
Proofs: 3 4

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  1. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994