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.
Best Answer
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 $N$.