1. What does it mean for a vector to form a basis of a subspace?
2. What does it mean when a plane (or subspace) is spanned by a vector?
3. What does it mean when a subspace has a basis?
- It means that the subspace it is in has dimension 1. Furthermore, the vector is nonzero every element in that subspace is a scalar multiple of that vector.
- Same as 1. Note that any nonzero vector is linearly independent by itself and so forms a basis for the subspace it spans.
- If you accept the axiom of choice (which most mathematicians do) – or rather, if you accept Zorn’s lemma, which is equivalent to the axiom of choice – then it follows that every vector space has a basis. A subspace of a vector space is itself a vector space and therefore has a basis.