basis of space and subspace 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.
 A: Yes, that's exactly right. Some set of vectors is a "basis" for V if those vectors are linearly independent and span V. Informally, "spanning" means that V is the smallest vector space that contains all of those vectors; "linearly independent" means that there are no redundant vectors (i.e. if you take one out, the new set of vectors spans a strictly smaller space).
Of course, linearly independent vectors will stay linearly independent regardless of the ambient space you consider them in. But vectors that span a space W will not necessarily span a space V. The ambient space matters. (If they span W, then they cannot span V, unless V = W.)
It might help you to try to work out which subspace of $\mathbb{R}^3$ the vectors $\begin{pmatrix}1\\0\\1\end{pmatrix}, \begin{pmatrix}2\\0\\3\end{pmatrix}$ span. Then show that they are linearly independent, to show that they are a basis of this subspace. Once you've worked that out, try to work out why can't they be a basis of anything smaller, bigger, or different.
