Span of 2 groups V is a linear space, Given that K and T are sub sets of V.
Does it mean that $Sp(K) + Sp(T) = Sp(K + T)?
$
I'm trying to find a way to prove it, but I've failed untl now, can anyone give a hint please?
I thought that if I do that:
$K = (a_1,...,a_n)$
$T = (b_1,...,a_n)$
and $SpK = (Q_1a_1,...,Q_na_n)
SpT = (W_1b_1,...,W_na_n)$
Then $$Sp(K) + Sp(T) = (Q_1a_1+W_1b_1,...,Q_naa_n+W_nb_n)$$
Which makes it true $Sp(K) + Sp(T) = Sp(K + T)$
 A: *

*You call $K$ a subset of $V$, but you write $K=(a_1,\dots,a_n)$, which looks more like the notation for a single vector than for a set of vectors. I think (but I'm not sure) that what you mean is $K=\{a_1,\dots,a_n\}$, where $a_1,\dots,a_n$ are elements of $V$. 

*You write that the span of $K$ is $(Q_1a_1,\dots,Q_na_n)$, which isn't right under any interpretation of the notation. Assuming that what you wanted for $K$ was $K=\{a_1,\dots,a_n\}$, as I wrote above, then the span of $K$ is the set of all elements of the form $Q_1a_1+\cdots+Q_na_n$ as the $Q_i$ run independently through the reals. 

*Your formula for the sum of the span of $K$ and the span of $T$ is the correct formula for the sum of two vectors, but since $K$ and $T$ are not vectors, but sets of vectors, your formula for their sum does not apply. 
Given all this, I'm not sure how to begin to write an answer to your question, but here goes: let $v$ be a nonzero element of $V$, let $K=\{\,v\,\}$, let $T=\{\,-v\,\}$. Then the span of $K$, the span of $T$, and the sum of the two spans are all the same; each is the set of all real multiples of $v$. But $K+T=\{\,0\,\}$, so the span of $K+T$ is $\{\,0\,\}$. 
