# 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 $\Im(z)=\Im(z^\prime).$

Proofs: 1

Proofs: 1

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### References

#### Bibliography

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

#### Footnotes

1. Please note that we consider $$\alpha,\beta$$ to be real numbers (scalars in the vector space), while we consider the numbers $$1$$ and $$i$$ as complex numbers (or vectors of the vector space).