Union of vectorspaces is never a vector space

Question

I know that the union of two different subvectorspaces $U_1,U_2$ of a vectorspace $V$ is in general not a vektor space.

But now I read

If $U_i$ $(i=1,2)$ has dimension $1$ and $V$ has dimension $2$, the union is never a vectorspace.

Does anyone know why?

by "union" do you mean $U_1\cup U_2 \cup V$? and are $U_i$ subspaces of $V$ here? — user160738, Dec 06 '16 at 15:45
But with $U_1 \ne U_2$, I suppose? Otherwise... (note that "never" is quite strong). — StackTD, Dec 06 '16 at 15:48

Vladimir Vargas · Accepted Answer · 2016-12-06T16:12:04.603

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This is the case of $V$ being a plane, and $U_i$ representing lines that pass through the origin. Clearly, if you take two vectors from the lines and you add them you will end up with a vector outside the lines, meaning that this is not a vector space.

More formally:

$U_1=\{\alpha(u^{(1)}_1,u^{(2)}_1), \alpha\in\mathbb{R}\}$ and $U_2=\{\alpha(u^{(1)}_2,u^{(2)}_2), \alpha\in\mathbb{R}\}$. Let $v =(v^{(1)},v^{(2)})=\alpha(u^{(1)}_1,u^{(2)}_1) + \beta(u^{(1)}_2,u^{(2)}_2)$ for some $\alpha,\beta\in\mathbb R$. If the union were to be a vector space, then there must exist $\gamma \in\mathbb R$ such that $v = \gamma(u^{(1)}_1,u^{(2)}_1)$ or $v = \gamma(u^{(1)}_2,u^{(2)}_2)$.

Solve each of the equations for $\gamma$ and convince yourself that there is no $\gamma\in\mathbb R$ that can can solve them. Therefore the assumption is false and the union cannot be a vector space.

edited Dec 06 '16 at 16:12

answered Dec 06 '16 at 15:46

Vladimir Vargas

4,635

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Not if the lines coincide. Also you might need to explain that $2$-dim vector spaces are isomorphic to $\mathbb{R}^2$ – user160738 Dec 06 '16 at 15:47
Yes, the lines don't coincide, otherwise it would be trivial and would contradict what we want to prove. – Vladimir Vargas Dec 06 '16 at 15:48
So this is a geometrically reason which is clear to me, thank to that! But I'd prefer a formal proof. – Dec 06 '16 at 15:52
@user114179 Please read the more formally written answer. Geometrical reasons are quite formal. If you understand the idea, you can write it as formal as you like, but that's your job. – Vladimir Vargas Dec 06 '16 at 16:13

score 0 · Answer 2 · edited Apr 13 '17 at 12:19

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The union of two subspaces can only be a subspace if one of the subspaces is contained in the other one (e.g. take a look here).

Since both subspaces are 1-dimensional and different, one can't be contained in the other (why?).

edited Apr 13 '17 at 12:19

Community

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answered Dec 06 '16 at 15:55

StackTD

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Union of vectorspaces is never a vector space

2 Answers2