Question

Show that if S1 and S2 are subsets of a vector space V such that S1 c S2 then span(S1) c span(S2). In particular, if S1 c S2 then Span(S1)

Answer 1

Answer:

See proof below

Step-by-step explanation:

Assume that V is a vector space over the field F (take F=R,C if you prefer).

Let $x\in span(S_1)$. Then, we can write x as a linear combination of elements of s1, that is, there exist $v_1,v_2,\cdots,v_k \in S_1$ and $a_1,a_2,\cdots,a_k\in F$ such that $x=a_1v_1+a_2v_2+\cdots+a_kv_k$. Now, $S_1\subseteq S_2$ then for all $y\in S_1$ we have that $y\in S_2$. In particular, taking $y=v_j$ with $j=1,2,\cdots,k$ we have that $v_j\in S_2$. Then, x is a linear combination of vectors in S2, therefore $x\in span(S_2)$. We conclude that $span(S_1)\subseteq span(S_2)$.

If, additionally  $S_2\subseteq S_1$ then reversing the roles of S1 and S2 in the previous proof, $span(S_2)\subseteq span(S_1)$. Then $span(S_1)\subseteq span(S_2)\subseteq span(S_1)$, therefore $span(S_1)=span(S_2)$.