Show that the set $B$ is a basis for vector space $V$

Question

I was reading a proof for rank-nullity theorem and a set $B$ was introduced. The goal was to show $B$ is a basis for vector space $V$. First, it was shown every vector in $V$ can be written as a linear combination of vectors in $B$ and then $B$ is a set of linear independent vectors. I wonder that is there a need for prove every linear combination of vectors in $B$ lies in $V$? I thought it is necessary but as I searched over other problems, regarding to show a set is a basis, there was no proof.

Answer 1

As $V$ is a vector space, it is closed under scalar multiplication and addition of elements (a fundamental property of vector spaces) and consequently also under linear combinations of elements. Since most likely $B\subseteq V$, you get the closure of $V$ under linear combinations of elements of $B$ for free.