Based on 2 sets of vectors in $\mathbb{R}^4$, how do I determine a system of equations? (Linear Algebra) Please help?

Consider the following 2 sets of vectors in $\mathbb{R}^4$: $A = \{v_1, v_2, v_3\}, B = \{w_1, w_2, v_3\}$. You are given that $A$ is a set of linearly independent vectors and that $B$ is a set of linearly independent vectors.

The intersection of 2 sets is the set of elements that are common to both sets. Suppose $u$ is in the intersection of span $A$ and span $B$. Determine a system of equations that could be used to determine all such $u$.

Please show steps and answers so that I can learn. Thank you so much.

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Can you write down the definition of a vector $u$ lying in $\operatorname{span} A$? Also, $\operatorname{span} B$? There's your system. –  Sammy Black Mar 26 '13 at 5:57

A similar calculation will yield real numbers $d_1,d_2,d_3,d_4$ such that $u$ is in the span of $B$ exactly when $u$ is a solution to the single equation $d_1\u1 + d_2\u2 + d_3\u3 + d_4\u4=0$. Therefore, detecting whether $u$ is in the intersection of the spans of $A$ and $B$ is equivalent to deciding whether $u$ is a solution to the pair of equations \begin{align*} c_1\u1 + c_2\u2 + c_3\u3 + c_4\u4&=0 \\\ d_1\u1 + d_2\u2 + d_3\u3 + d_4\u4&=0. \end{align*}