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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
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1 Answer

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*}

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