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Assume that $x$ is an n-dimensional vector of complex values, namely $x= x_{re} + ix_{im}$ where both $x_{re}$ and $x_{im}$ are n-dimensional real column vectors.

Assume that we restrict $x_{im}$ to be always the same vector, while $x_{re}$ can be any real vector.

What are the properties of corresponding restricted complex vector space? Does such a restricted complex vector space have significance? Is there a mathematical notion describing such a situation?

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I'll notate with tuples for the simplicity, but we're still talking about the same thing. At any rate, you'd have something like $\{(x_1+ai, ..., x_n+ai): a,x_1,...,x_n \in \mathbb{R})\}$. We'll call this $A$.

The most notable property of this is that it is, in fact, not a vector space. Recall our requirements (here is a nicely done list that was at the top of my Google results).

If we ever have $a$ (that is, $x_{im}$) non-zero, we run into some issues. I'll leave this to you, but will tag on that it's the same reason as to why the empty set isn't a vector space (at least, that's probably the lowest-hanging fruit; it breaks more than just one of our criteria).

If $a\neq 0$, then $0\notin A$.

If $a\neq 0$, it is not closed under addition nor scalar multiplication. Consider $u+v$ for $u,v \in A$. Any one coordinate would have complex part $a$, but $u+v$ would have complex part $2a$, thereby meaning that $u+v\notin A$. Scalar multiplication is similar.

I'll drop an et cetera here. Your defined set of vectors does not only break those three axioms alone.

As to a mathematical notion for this: we'd need $a=0$ for this to be a vector space, but, at that point, we'd just have $\mathbb{R}^n$ with some extra steps. The remaining pieces of your questions were rather directly tied to this being a vector space; delving into such discussion in the terms of vector spaces isn't really bringing us much, though the notion is interesting! That being said, this math.SE post may be of interest as well. This one is probably of value to link as well. Thirdly, you may find abstract algebra interesting if you like exploring questions like these; the field deals with such structures.

Edit: a second thought that might be somewhat more insightful.

Consider constructing this proposed set of vectors as the span of a vector or two. What would those vectors be here? Definitely tough to put together (and really, you can't). Fix $n$. Pick a set of $n$ basis vectors to span our proposed vector space. ...

Where does this fall apart?

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  • $\begingroup$ Yes you are right that this is not a vector space. However, we can still change or tweak addition and multiplication operations, can't we? Perhaps it still will not be a vector space, but at least we may sustain closure somehow. Your last thought sounds promising also. $\endgroup$
    – user
    Commented Jan 22, 2021 at 8:39
  • $\begingroup$ If we were to tweak our definitions of multiplication and addition, it would leave us still without a vector space; think about adding two complex numbers and where that puts us geometrically. I'm assuming you're meaning to leave out the complex part or something like that: $ca+cbi$ sends us to a place very different from $ca+bi$. The last thought was to put a more geometric intuition behind this. Every vector space can be written as the span of vectors, and every span of vectors is a vector space. $\endgroup$
    – user838358
    Commented Jan 22, 2021 at 10:31

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