Vector Space over Division Subring/Examples/Real Numbers in Complex Numbers

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Example of Vector Space over Division Subring

Consider the field of complex numbers $\struct {\C, +, \times}$, which is a ring with unity whose unity is $1$.

Consider the field of real numbers $\struct {\R, +, \times}$, which is a division subring of $\struct {\C, +, \circ}$ such that $1 \in \R$.

Then $\struct {\C, +, \times_\R}_\R$ is an $\R$-vector space, where $\times_\R$ is the restriction of $\times$ to $\R \times \C$.

$\struct {\C, +, \times_\R}_\R$ is of dimension $2$.

The set $\set {1 + 0 i, 0 + i}$ forms a basis of $\struct {\C, +, \times_\R}_\R$, as do any two complex numbers which are not real multiples of each other.