Is it possible for a set of non spanning vectors to be independent?

Question

I was reading about linear spans on Wikipedia and they gave examples of spanning sets of vectors that were both independent and dependent. They also gave examples of non spanning sets of vectors that are dependent. My question is whether it is possible for a set of non spanning vectors to be independent when the number of vectors in the set is equal to the number of dimensions?

For example, if we have the following set in $R^3$, {(1,2,0), (2,3,0), (5,7,0)}, then the vectors do not span $R^3$ and are not independent. Based on this example I have a feeling that it is not possible to for a set of non spanning vectors to be independent but I was looking for a more rigorous proof. Ideas?

Answer 1

It is possible. The vectors (1,0,0) and (0,1,0) are linearly independent but do not span $\mathbb{R}^{3}$. Any subset of a basis of a vector space will be linearly independent (because, by definition, the whole basis set is and the subset still satisfies the conditions for linear independence), but will not span the original whole vector space (since, by definition, the basis is the minimal spanning set), although your new subset will span a subspace thereof.

Answer 2

Take any basis of a finite-dimensional space, and remove one of the vectors. This will remain independent (subsets of independent sets of vectors are independent themselves), but it will no longer span the space.

Answer 3

You are very close to being right. It is correct that if you have three vectors in $\mathbb R^3$ which do not span $\mathbb R^3$, then they are necessarily dependent. This follows from the dimension theorem, since if they were independent, they would be a proper subset of a basis, which would have more elements than the dimension of the space, a contradiction.