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

See proof below

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)`.