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[Community Question] Linear-algebra: Which subspaces of $\mathbb C^n$ are spanned by real vectors?

One of our user asked:

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 $W=\text{span}_{\mathbb C}(v_1,\ldots,v_k)$

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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