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.