In Linear Algebra by Friedberg, Insel and Spence, the definition of span (pg-$30$) is given as:
Let $S$ be a nonempty subset of a vector space $V$. The span of $S$,
denoted by span$(S)$, is the set containing of all linear
combinations of vectors in $S$. For convenience, we define
span$(\emptyset)=\{0\}$.
In Linear Algebra by Hoffman and Kunze, the definition of span (pg-$36$) is given as:
Let $S$ be a set of vectors in a vector space $V$. The subspace
spanned by $S$ is defined to be intersection $W$ of all subspaces of
$V$ which contain $S$. When $S$ is finite set of vectors, $S =
\{\alpha_1, \alpha_2, …, \alpha_n \}$, we shall simply call $W$ the
subspace spanned by the vectors $\alpha_1, \alpha_2, …, \alpha_n$.
I am not able to understand the second definition completely. How do I relate "set of all linear combinations" and "intersection $W$ of all subspaces"? Please help.
Thanks.
Best Answer
Remember that a subspace by definition is closed with respect to vector addition. That means that every subspace which contains $S$ necessarily contains every linear combination of elements of $S$. In turn then, the intersection of all such subspaces is exactly the set of all linear combinations of vectors in $S$.