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

If I take an element $ v$ in $ (<U_1> \cap <U_2>)$ 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 $ (<U_1> \cap <U_2>)$ must be a linear combination of vectors that are bothin $ U_1$ and $ U_2$