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. 
 A: Let's first think about how we could figure out whether $u$ was in the span of $A$. We would need to see if the vector equation $a_1v_1 + a_2v_2 + a_3v_3 = u$ had a solution $(a_1,a_2,a_3)$ in real numbers. To decide whether such a solution exists, we'd write down the corresponding augmented matrix
$$
\newcommand{\v}[2]{v_{#1}^{(#2)}}
\newcommand{\u}[1]{u^{(#1)}}
\begin{pmatrix}
\v11 & \v21 & \v31 & | & \u1 \\\
\v12 & \v22 & \v32 & | & \u2 \\\
\v13 & \v23 & \v33 & | & \u3 \\\
\v14 & \v24 & \v34 & | & \u4
\end{pmatrix}
$$
(here $\u1,\u2,\u3,\u4$ are the components of the vector $u$, and similarly for the $v_i$), and then reduce the coefficient matrix to row-echelon form, yielding something of the form
$$
\begin{pmatrix}
1 & * & * & | & * \\\
0 & 1 & * & | & * \\\
0 & 0 & 1 & | & * \\\
0 & 0 & 0 & | & c_1\u1 + c_2\u2 + c_3\u3 + c_4\u4
\end{pmatrix}
$$
(where $c_1,c_2,c_3,c_4$ come up during the row operations used). This system is consistent exactly when $c_1\u1 + c_2\u2 + c_3\u3 + c_4\u4=0$. In other words, $u$ is in the span of $A$ exactly when $u$ is a solution to the single equation $c_1\u1 + c_2\u2 + c_3\u3 + c_4\u4=0$.
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*}
