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My instructor for Linear Algebra gave us a problem to think about but am quite unsure on how to approach it:

Let $V_1, V_2, ... V_{100}$ be $100$ proper subspaces of the complex vector space $V=\mathbb C^2$. Can it be possible that $\bigcup _{i=1}^{100} V_i = \mathbb C^2$?

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marked as duplicate by Martin, Dominic Michaelis, Davide Giraudo, TMM, Henry T. Horton Mar 31 '13 at 17:15

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What are the possible subspaces of $\mathbb{C}^2$? $\{0\}$, $\{z(\alpha_1,\alpha_2): z \in \mathbb{C}\}$ and what else? –  Pedro M. Mar 31 '13 at 15:56
    
Also related: this and this. –  Martin Mar 31 '13 at 16:52

3 Answers 3

Nope, there are two cases, all $V_i$ are the same, then the union is a subspace to, and a proper one. The second case is that not all $V_i$ are the same, then the union of all $V_i$ is not closed under addition and can't be a vector space. I used here that we have a 2 dimensional vectorspace over an infinite field, so we have an infinite number of subspaces.

Proof the following Lemma:
Let $U_1,U_2$ be subspaces of a vectorspace $V$, then \[ U_1 \cup U_2 \] is a subspace iff $U_1\subseteq U_2$ or $U_1 \supseteq U_2$

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Allow me to disagree. Of course the union of two subspaces is a subspace if and only if it coincides with one of them. But if you consider the union of a finite number, greater than two, of subspaces in a finite vector space, this can be a subspace different from each of them. So any argument here has to use the fact that we are working over an infinite field. –  Andreas Caranti Mar 31 '13 at 16:32
    
@AndreasCaranti I implicitly used that a 2 dimensional vector space over an infinite field does have infinite many proper subspaces, if you only have finitely many proper subspaces you are right. –  Dominic Michaelis Mar 31 '13 at 16:38
    
@AndreasCaranti I think I fixed it now, at downvoter is there still something to fix ? –  Dominic Michaelis Mar 31 '13 at 16:56
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I have never given a downvote in all my MSE life. –  Andreas Caranti Mar 31 '13 at 16:57
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+1 Same for me, @Andreas: comments are so much better... –  Georges Elencwajg Mar 31 '13 at 17:11

If $V_i$ has as equation $l_i(z,w)=a_iz+b_iw=0$, then the non-zero polynomial $P(z,w)=\prod^{100}_{i=1} l_i(z,w)$ of degree $100$ vanishes on $\bigcup^{100}_{i=1} V_i$.
Since a non-zero polynomial cannot vanish on the whole of $\mathbb C^2$, we have $\bigcup^{100}_{i=1} V_i\subsetneq \mathbb C^2$

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One has to be careful, because a finite vector space can well be the union of finitely many proper subspaces. For instance the vector space of dimension two over the field with two elements is the union of three subspaces of dimension one.

Note first that each $V_i$ can be assumed to be one-dimensional. Add the subspace $V_{0} = \{ (0, a) : a \in \Bbb{C} \}$ to the $V_{i}$ for good measure, and suppose each $V_i \ne V_0$ for $i \ne 0$.

Then in each $V_i$ (with $i \ne 0$) there is a unique element of the form $(1, a_i)$. Choose an element $a \notin \{ a_i : i \in 1, \dots, 100 \}$. (Here of course we are using the fact that $\Bbb{C}$ is infinite.) Then $(1, a) \notin \bigcup_{i=0}^{100} V_i$, so $V \ne \bigcup_{i=0}^{100} V_i$.

So this works for any finite number of subspaces, and any infinite field.

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