In Linear Algebra by Friedberg, Insel and Spence, the definition of span (pg-3030) is given as:
Let SS be a nonempty subset of a vector space VV. The span of SS,
denoted by span(S)(S), is the set containing of all linear
combinations of vectors in SS. For convenience, we define
span(∅)={0}(∅)={0}.
In Linear Algebra by Hoffman and Kunze, the definition of span (pg-3636) is given as:
Let SS be a set of vectors in a vector space VV. The subspace
spanned by SS is defined to be intersection WW of all subspaces of
VV which contain SS. When SS is finite set of vectors, S={α1,α2,...,αn}S={α1,α2,...,αn}, we shall simply call WW the
subspace spanned by the vectors α1,α2,...,αnα1,α2,...,αn.
I am not able to understand the second definition completely. How do I relate "set of all linear combinations" and "intersection WW of all subspaces"? Please help.
Thanks.
