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

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.

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  • $\begingroup$ 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). $\endgroup$ – Mees de Vries Jan 29 at 13:19
  • $\begingroup$ 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". $\endgroup$ – Asaf Shachar Jan 29 at 13:53
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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 $\Im(w_i)$.

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  • $\begingroup$ 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. $\endgroup$ – Asaf Shachar Jan 30 at 9:54
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    $\begingroup$ @AsafShachar If we consider the real and imaginary parts as complex vectors, they form a spanning set with 2k vectors for a k-dimensional space, so k of them must be a basis and suffice to span W (in particular, the other k vectors, giving relations). Does this help, or is there a different problem? $\endgroup$ – Jose Brox Jan 30 at 10:57
  • $\begingroup$ No, this is perfect. Thank you for this nice answer. $\endgroup$ – Asaf Shachar Jan 30 at 14:06

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