What is the “standard basis” for fields of complex numbers?

What is the “standard basis” for fields of complex numbers?

For example, what is the standard basis for C2 (two-tuples of the form: (a+bi,c+di))? I know the standard for R2 is ((1,0),(0,1)). Is the standard basis exactly the same for complex numbers?

P.S. – I realize this question is very simplistic, but I couldn’t find an authoritative answer online.

Answer

Just to be clear, by definition, a vector space always comes along with a field of scalars F. It’s common just to talk about a “vector space” and a “basis”; but if there is possible doubt about the field of scalars, it’s better to talk about a “vector space over F” and a “basis over F” (or an “F-vector space” and an “F-basis”).

Your example, C2, is a 2-dimensional vector space over C, and the simplest choice of a C-basis is {(1,0),(0,1)}.

However, C2 is also a vector space over R. When we view C2 as an R-vector space, it has dimension 4, and the simplest choice of an R-basis is {(1,0),(i,0),(0,1),(0,i)}.

Here’s another intersting example, though I’m pretty sure it’s not what you were asking about:

We can view C2 as a vector space over Q. (You can work through the definition of a vector space to prove this is true.) As a Q-vector space, C2 is infinite-dimensional, and you can’t write down any nice basis. (The existence of the Q-basis depends on the axiom of choice.)

Attribution
Source : Link , Question Author : Casey Patton , Answer Author : Jonas Kibelbek

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