If a subset of $S$, say $R$ is a basis for vector space $V$, then is $S$ also a basis for $V$?

Question

If a subset of $S$, say $R$ is a basis for vector space $V$, then is $S$ also a basis for $V$? I want to say yes, but I am not $100\% $ sure. If it is, then can I say that if $R$ has $n$ vectors in it, $S$ has at least n vectors too?

Answer 1

If a subset of $S$, say $R$ is a basis for vector space $V$, then is $S$ also a basis for $V$?

No, because if you add any element to the basis, the resulting set is linearly dependent. A counterexample to your claim: Let $V = \mathbb{R}^2$ and $S = \{(1,0)^T, (0,1)^T, (0,2)^T \}$ and $R = \{(1,0)^T, (0,1)^T \}$. Then $R$ is a basis for $V$ but $S$ is not because $S$ is linearly dependent.

What you can say is that $S$ will span $V$ if $R$ spans $V$, but as in the example, the vectors of $S$ will not be linearly independent, and therefore not a basis.

Answer 2

Just take $R^3$ and standard basis. If you add any vector to this set, you will have a linear dependent set which is not a basis, but standard basis is in this set(as a subset). But if you have a linear independent set, you can always extend it to get a basis. Converse is true for a set which generates the space : you can always find a subset which is a basis.