V is a vector space over a field F and A is a subspace. $\displaystyle x,y\in V$ have two lin. ind. combinations which are in A. Show $\displaystyle x,y\in A$

Let $\displaystyle \lambda_i\in F$

$\displaystyle x=\lambda_1v_1+\cdots +\lambda_nv_n\in A$

$\displaystyle y=\lambda_1v_1+\cdots +\lambda_mvm\in A$

Since x = the linear combinations in A, x is in A. Is it really this simple?