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.
Best Answer
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 $\Im(w_i)$.