# Vector subspaces are the same as the parent vector space.

I came across this question in Halmos's book which i was not sure how to answer

"If M and N are subspaces of a vector space V, and if every vector V belongs either to M or to N(or both), then either M = V or N = V or both."

As i was told inorder to show two finite vector spaces are the same it is enough to show basis of the first is also the basis of the second.

Proof Attempt Assuming m is basis of M and n is a basis of N and based on the information in the question that every vector of V belongs to M or to N (or both), then either the dim V = dim M or dimV = dim N or both must hold otherwise the it would not be possible for every vector of V to belongs to M or to N (or both). If the dimensions of the basis are the same then to span the whole vector space all we need to do is find the appropriate scalars to represent any vector in V. Hence we conclude M = V or N = V or both.

Was my reasoning and proof correct ? If not how to improve it. Any guidance or help would be much appreciated.

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What you've written isn't a proof. In the end of your second sentence you're basically appealing to the statement you're trying to prove. I would suggest that you try to prove the contrapositive. –  Qiaochu Yuan Feb 7 '12 at 23:49
Cool thanks very much i would try that. –  Hardy Feb 7 '12 at 23:51
@Hardy Related problem. Suppose that $U$ is a subspace of $V$ and $\operatorname{dim} U = \operatorname{dim} V$. Prove that $U = V$. –  user38268 Feb 8 '12 at 0:00
I suppose i'd try to form an argument based on the contradiction let's assume U is not equal to V, so that implies there are vectors in V which are not present in U. Since they both have the same dimensions and and U is a subspace of V they must have some common vectors and that would require them to have the same basis, which would imply that the only way there could be vectors in V which are not in U if there is a restrictions on the scalar values allowed for U, as in a different field is used which woudl violate the definition of the vector space. what do u think ? –  Hardy Feb 8 '12 at 0:14

If both $M\ne V$ and $N\ne V$ then you can find $v\notin M$ and $w\notin N$. The vector $v+w$ can not belong to $M$ or $N$.