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?

  • 1
    $\begingroup$ 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 $s\cup v$? $\endgroup$ – EuYu Mar 24 '13 at 9:09
  • $\begingroup$ Do you mean span {$s$}$\cup${$u$}=Span{$s$}+span{$v$}? $\endgroup$ – jim Mar 24 '13 at 9:12
  • $\begingroup$ Hey i edited the question. Hope it is clear now. $\endgroup$ – 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.


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