What is the Real Coordinate Space in the discussion of vectors?

How does it relate to Cartesian Coordinate System and Euclidean Space?

P.S. Please, use naive terms.

up vote 1 down vote accepted

In my experience the expression "real coordinate space" emphasizes that we are not working over the complex numbers, i.e. the space is $\mathbb R^n$, not $\mathbb C^n$. You can use Cartesian coordinates (and a whole bunch of other coordinate systems) on these spaces. The spaces $\mathbb R^n$ are called Euclidean spaces, so they are the same as real coordinate spaces.

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