Which subspaces of $\mathbb C^n$ are spanned by real vectors?

Which complex $$k$$-dimensional subspaces of $$\mathbb C^n$$ are spanned by real vectors? Can we characterise them? (here $$1).

By "complex", I mean that I am interested in subspaces $$W \le \mathbb C^n$$, which admit $$k$$ vectors $$v_1,\ldots,v_k \in \mathbb{R}^n$$, such that $$W=\text{span}_{\mathbb C}(v_1,\ldots,v_k)$$.

This is equivalent to $$W \cap \mathbb R^n$$ being a $$k$$-dimensional real vector space. More explicitly, suppose $$W$$ is such a subspace, and that $$W=\text{span}_{\mathbb C}(w_1,\ldots,w_k)$$ for some $$w_i \in \mathbb C^n$$. Are there some relations the $$w_i$$ must satisfy?

Of course, the $$w_i$$ themselves do not have to be real, since we can start with a real spanning set, and multiply some of its elements by $$i$$.

In the case $$n=2,k=1$$, we ask when $$(z_1,z_2)$$ can be expressed as $$z_0\cdot(x_1,x_2)$$ for some $$z_0 \in \mathbb C$$ and $$x_1,x_2 \in \mathbb R$$. This is equivalent to $$z_1$$ being a real multiple of $$z_2$$ or vice versa.

• What sort of answer are you looking for? The question is equivalent to asking whether $W_{\mathbb R} = \{w \in W \mid w \in \mathbb R^n\}$ is $k$-dimensional (as a real vector space). – Mees de Vries Jan 29 at 13:19
• Hmmm, you are right. I am not really sure. My vague thinking was something like specifying some relations on a generating set. Indeed, I should add the tag "soft question". – Asaf Shachar Jan 29 at 13:53

We have that $$W=$$span$$_{\mathbb{C}}\{w_1,\ldots,w_k\}$$ with $$w_i\in\mathbb{C}^n$$ is spanned by $$\{v_1,\ldots,v_k\}\subseteq\mathbb{R}^n$$ if and only if $$\Re(w_i),\Im(w_i)\in W$$ for all $$i$$, with the real and imaginary maps understood componentwise.
This is clear: If the condition is satisfied then $$W$$ is spanned by the real and imaginary parts of the $$w_i$$. On the other hand, if $$w_i=\sum_i \lambda_i v_i$$ then $$\Re(w_i)=\Re(\sum_i \lambda_i v_i) = \sum_i \Re(\lambda_i) v_i$$ is in $$W$$ and the same for $$\Im(w_i)$$.
• Thanks. I am not sure I understand the first direction: If the imaginary and real components of each $w_i$ are in $W$, don't you need both of them to span $W$? This only implies that $W$ can be spanned by $2k$ real vectors instead of $k$ vectors... I guess that I am missing something. – Asaf Shachar Jan 30 at 9:54