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I have a question which is pretty basic about basis and verctor spaces.
Generally, if I have a basis K of vector space V, Why it is not a basis of a subspace W of V (real subspace)?
The vectors in basis K are linearily independent and for every vector of W I can find a linear combination of the vectors in the basis K that equals to the vector in W.
The only problem I can see is that when i make some linear combinations of the vectors in basis K , I get vectors that belong to V but does not belong to the space W, is this the problem here? When I get vectors beyond my space that means it is not the proper basis , because it is spanning more vectors than my space? does it have to span exactly the group and not beyond? Thank you very much.
Sorry for my english and I don't know how to edit the question to be pleasant to read.