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 \(\alpha=\beta=0\).
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). ↩