◀ ▲ ▶Branches / Number-systems-arithmetics / Lemma: Linear Independence of the Imaginary Unit `\(i\)` and the Complex Number `\(1\)`

Lemma: Linear Independence of the Imaginary Unit \(i\) and the Complex Number \(1\)

Given the fact that complex numbers form a vector space over the field of real numbers, the vectors * complex number one \(1\) and the * imaginary unit \(i\)

are linearly independent, i.e. for some real numbers¹ \(\alpha,\beta\in\mathbb R\), the equation \[\alpha\cdot 1 + \beta \cdot i=0\] has only the trivial solution \(\alpha=\beta=0\).

Notes

• This lemma demonstrates that two complex numbers $z$ and $z^\prime$ are equal if and only if their real and imaginary parts are equal $\Re(z)=\Re(z^\prime)$ and $\Im(z)=\Im(z^\prime).$

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1

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References

Bibliography

1. Timmann, Steffen: "Repetitorium der Funktionentheorie", Binomi-Verlag, 2003

Footnotes