Generating a vector space
Given a vector space V over a field K, a subset W of V is said to generate (or span) V if every element of V can be expressed as a linear combination of elements of W.
Example 1:. Let U and W be subspaces of the vector space V over K.
Define U+W={ u+v | u is in U and w is in W }. Then U+W, called the sum of U and W, is a vector subspace of V and is spanned by the union of U and W.
Proof:
Property 1: Let v1 and v2 belong to U+W.
Then v1 = u1 + w1
and v2 = u2 + w2,
for some u1 and u2 in U and some w1 and w2 in W.
Then v1 + v2 = (u1 + w1) + (u2 + w2) = (u1 + u2) + (w1 + w2), which is a member of U+W.
Property 2: Let v be a member of U+W and c be a member of K. Consider
cv. Since v is a member of U+W, v=u+w for some u in U and w in W.
Then cv = c(u+w)=cu+cw. Since U and W are themselves vector spaces, cu is a member of U and cw is a member of W. So cv is a member of U+W.
Property 3: Since U and W are vector spaces, 0 is a member of both U
and W. Then 0+0 = 0 is a member of U+W.
Any element u+w of U+W is clearly a linear combination of an element from U and an element from W, so U+W is generated by the union of U and W.
Example 2: Let S be a subspace of R3, defined by S =
{ (s1,s2,s3) | s2=0 }. Then S is generated by e1 = (1,0,0) and e3 = (0,0,1).
Proof:
Let v belong to S. Then v = (v1,0,v3) for some real numbers v1 and v3. To show that V is generated by e1 and e3, it is necessary to prove that v is a linear combination of e1 and e3. But v1e1+v3e3 = (v1,0,0)+(0,0,v3) = (v1,0,v3) = v, as required.