Let u and v be any vectors in a vector space V.

Prove that for any vectors xsub1 and xsub2 in V, span{xsub1, xsub2} is a subset of span { u, v} if and only if xsub1 and xsub2 are linear combinations of u and v.

There really isn't anything to prove here - this is just looking at the definitions and matching things up. For example, for proof of linear combination implies subset:

 

[math]w \in \text{span}\{x_1, x_2\} \Rightarrow w = \alpha_1 x_1 + \alpha_2 x_2; \alpha_i \in K \Rightarrow w = \alpha_1 (\beta_1 u + \beta_2 v) + \alpha_2 (\gamma_1 u + \gamma_2 v ); \beta_i, \gamma_i \in K[/math]

 

Hence by letting [math]a_i = \alpha_1\beta_i + \alpha_2\gamma_i \in K[/math],

 

[math]w = a_1 u + a_2 v \Rightarrow w \in \text{span}\{ u, v \} \Rightarrow \text{span}\{x_1,x_2\} \subset \text{span}\{u,v\}[/math]