# $span\{s+v\} = span\{s\} + span\{v\}$?

span{s+v} = span{s} + span{v}

Does the above statement always hold true?

In addition, what is the difference between span {s} $\cup$ {v} = span{s} + span{v} and the above statement?

• Are $s$ and $v$ vectors or sets of vectors? If they are sets, how are you defining $s+v$? If they are vectors, how are you defining $s\cup v$? – EuYu Mar 24 '13 at 9:09
• Do you mean span {$s$}$\cup${$u$}=Span{$s$}+span{$v$}? – jim Mar 24 '13 at 9:12
• Hey i edited the question. Hope it is clear now. – uohzxela Mar 24 '13 at 9:17

No. Consider the set {$(1,0)+(0,1)$} $=$ {$(1,1)$}.
The span of {$(1,1)$} is just a straight line through the origin in $\mathbb{R}^{2}.$ While the span of {$(0,1)$,$(1,0)$} is all of $\mathbb{R}^{2}$.
On the other hand span({$s$} $\cup$ {$v$}) $=$ span{$s$} + span{$v$} holds.