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Does saying "V is a vector space over reals" same as real vector space? For eg:

$$ V={M}_{2}\left( {\mathbb{C}} \right)\\ M_2 \text{ is a } 2 \times 2 \text{ matrix and } \mathbb{C} \text{ is the field of complex numbers}\\ W= {\{\begin{bmatrix} a \; b \\ c \; d\end{bmatrix} | \; a= d' \}}$$ Here my book states that W is not a subspace of complex vector space $${M}_{2}\left( {\mathbb{C}} \right)$$but it is a subspace of real vector space $${M}_{2}\left( {\mathbb{C}} \right)$$ Does real vector space here means that a,b,c,d are real numbers but the chosen field for scalar multiplication is complex?
I am very confused. (here $\mathbb{C}$ is referring to the field of complex numbers)

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"Real" or "complex" here refers to the field of the scalars for your vector space.

Take the "real vector space $\mathbb{C}^1$", it's just the complex numbers with complex addition as the group law, and scaling understood as multiplication of complex numbers by a real numbers, but complex multiplication is forbidden (cause we're not in an $\mathbb{R}$-algebra, only in an $\mathbb{R}$-vector space). So it's really isomorphic to $\mathbb{R}^2$ in this case.

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  • $\begingroup$ You said complex multiplication is forbidden but while defining scalar multiplication of vector V by scalar A don't we say "A belongs to field F where F can be real or complex field"? Also could you explain in terms of the example (whether I am correct in the question or not)? $\endgroup$
    – Aditi Gupta
    Commented Mar 18, 2021 at 15:11
  • $\begingroup$ Sure, but there's a choice between the two, at the very start ! If you want to be coherent, you have to be really careful as to "what structure you're in". You can make a complex vector space over C, or a real vector space over C, or a real vector space over R, but not a complex vector space over R (since scalar multiplication by a number of C of a real coordinate in your vector could give you a non-real number, and thus would not be closed). So when you're making the choice of a "real vector space over C", your scalars are ONLY the real numbers. $\endgroup$
    – Tristan Duquesne
    Commented Mar 18, 2021 at 15:16
  • $\begingroup$ As for your example, what's your $a=d'$, precisely ? $d'$ doesn't seem defined $\endgroup$
    – Tristan Duquesne
    Commented Mar 18, 2021 at 15:17
  • $\begingroup$ Oh i see, so basically when we say "real/complex vector space over R/C", we mean that the vector will be constructed using R/C and real/complex defines the scalars used. (I used R/C and real/complex to show to which part of the phrase i was referring to) and in the example d' means complex conjugate but i think i understand now. Thank you! $\endgroup$
    – Aditi Gupta
    Commented Mar 18, 2021 at 16:09
  • $\begingroup$ Yes, it is precisely the distinction between "scalars" and "coordinates of vectors". A "real vector space over C" will have real numbers as scalars, and vectors with complex numbers as coordinates. $\endgroup$
    – Tristan Duquesne
    Commented Mar 18, 2021 at 16:16

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