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I was going through a question which had multiple correct answers. I was given a set $V$ and options were these

A) $V$ is a vector space over $\mathbb{R}$

B) $V$ is a vector space over $\mathbb{C}$

C) $V$ is not a vector space over $\mathbb{R}$

D) $V$ is not a vector space over $\mathbb{C}$

I am not posting the question because I don't want the solution of the particular problem. I just want to see a larger picture. I have two basic questions. First, if $V$ is a vector space over $\mathbb{C}$ then is it true that $V$ will always be a vector space over $\mathbb{R}$ ?

That is, what I mean is: does being a vector space over $\mathbb{C}$ always imply being a vector space over $\mathbb{R}$?

Second question: also, is it possible that if $V$ is not a vector space over $\mathbb{R}$ then it turns out in some case that $V$ is a vector space over $\mathbb{C}$?

That is, what I mean is: if $V$ is not a vector space over $\mathbb{R}$ can it be a vector space over $\mathbb{C}$?

symplectomorphic
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Abhishek Chandra
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  • If $V$ is a $\mathbb{C}$ vector space of dimension $n$ then it is a vector space of dimension $2n$ over $\mathbb{R}$. This is a consequence of the fact that $\mathbb{C}$ is a degree 2 field extension of $\mathbb{R}$. You can have $\mathbb{R}$ vector spaces which are not $\mathbb{C}$ vector spaces, say $\mathbb{R}^3$. – user2902293 Dec 29 '16 at 22:43
  • An interesting question is, whether $\mathbb{R}$ itself can be considered as a $\mathbb{C}$-vector space. For this question see here. – Dietrich Burde Dec 29 '16 at 22:47
  • It seems that your first and second questions are ultimately asking after the truth of the same thing. One asks "Is [blah] true?" The other asks "Is [blah] false?" – Cameron Buie Dec 30 '16 at 02:36

3 Answers3

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Any vector space over a field $L$ is, by restriction of scalars; a vector space over any subfield $K\subset L$. Hence the answer to your first question is ‘yes’. Furthermore, if $L$ is a finite extension of $K$ of degree $n=[L:K]$, then $$\dim_K v=\dim_L V\cdot [L:K].$$

Hence also the answer to your second question is ‘no’, since if $V$ were a vector space over $\mathbf C$, it would be ipso facto an $\mathbf R$-vector space.

Bernard
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If $V$ is a vector space over $\mathbb{C}$ then $V$ is always a vector space over $\mathbb{R}$.

Take a basis $\{e_1,e_2,\dots,e_n\}$ of $V$ as a vector space over $\mathbb{C}$. Consider the set of vectors $B=\{e_1,ie_1,e_2,ie_2,\dots,e_n,ie_n\}$. The set $W$ of linear combinations with real coefficients of elements of $B$ is clearly a vector space of dimension $2n$ over $\mathbb{R}$ and it is equal to $V$, because any $v\in V$ can be written as a linear combination: \begin{equation} v=z_1e_1+\cdots+z_n e_n \end{equation} with each $z_j=x_j+iy_j\in\mathbb{C}$ and any element of $w\in W$ can be uniquely written as a linear combination \begin{equation} w=x_1e_1+y_1(ie_1)+\cdots+x_ne_n+y_n(ie_n) \end{equation}

coconut
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For a field $K$ and its subfield $L\subseteq K$, if $V$ is a set of vectors, then

$V(K)$ is a vector space$\implies V(L)$ is a vector space.

The contrapositive of the above conditional is:

$V(L)$ is not a vector space $\implies V(K)$ is not a vector space.

which answers your second question.

Nitin Uniyal
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