If $U_1,U_2$ linear independent sets then $( \cap )=$

If I take an element $$v$$ in $$( \cap )$$ why this element can be described as :

$$v=\sum_{i=1}^{k}\lambda_iz_i+\sum_{i=k+1}^{k+n}\lambda_{i}x_i=\sum_{i=1}^{k}\mu_iz_i+\sum_{i=k+1}^{k+m}\mu_i y_i$$

?

where $$z_1,..,z_n\in U_1\cap U_2,x_{k+1},…,x_{k+n}\in U_1\backslash U_2$$ and $$y_{k+1},…,y_{k+m}\in U_2\backslash U_1$$

I thought every element of $$( \cap )$$ must be a linear combination of vectors that are bothin $$U_1$$ and $$U_2$$