In linear algebra, does span(G) = V means V is generated by G? In linear algebra, does span(G) = V means V is generated by G?
I have problem understanding the word "span" and "generating" in mathematic form.
 A: Presumably, $G$ is a collection of vectors in a vector space. By the span of $G$, we mean the set of vectors realizable as linear combinations of the vectors in $G$, which happens to be a vector space itself (we are calling this vector space $V$ here ).
Saying span$(G)=V$ means that every vector in $V$ can be written as a linear combination of vectors in $G$.
A: To say that a subset $G$ of a vector space $V$ generates $V$ means that $\text{span}(G) = V$.
A: In short: yes.
To generate $A$ from $B$ means to take (some or all) elements of $A$, use the allowed operations, and get all elements of $B$.
In case of a vector space, you take vectors in $G$, make all their linear combinations, and obtain a vector space. If you got $V$ (and not just its strict subspace), we say that the elements of $G$ generate $V$. Otherwise, they generate a strict subspace of $V$.
Span, denoted $\operatorname{span}(G)$, is the set (incidentally, the vector space) generated by all vectors in $G$. Think of the umbrella and its stretchers, and how they span the umbrella when you open it.
So, the elements of $G$ always generate $\operatorname{span}(G)$. If $\operatorname{span}(G) = V$, then they generate $V$.
