## Linear Algebra and Its Applications (5th Edition)

$w\not\in$Span$S$
Let $S=\{v_{1},v_{2},v_{3}\}$. If $w\in$Span$S$, then there exist $a,b,c\in \mathbb{R}$ such that $w=av_{1}+bv_{2}+cv_{3}.$ $a, b,$ and $c$ are solutions (if they exist) to a linear system with the augmented matrix $\left[\begin{array}{lllll} 1 & 2 & 4 & | & 8\\ 0 & 1 & 2 & | & 4\\ -1 & 3 & 6 & | & 7 \end{array}\right]\left\{\begin{array}{l} .\\ .\\ \leftarrow R_{3}+R_{1}. \end{array}\right.$ $\left[\begin{array}{lllll} 1 & 2 & 4 & | & 8\\ 0 & 1 & 2 & | & 4\\ 0 & 5 & 10 & | & 15 \end{array}\right]\left\{\begin{array}{l} .\\ .\\ \leftarrow (\frac{1}{5}R_{3}). \end{array}\right.$ $\left[\begin{array}{lllll} 1 & 2 & 4 & | & 8\\ 0 & 1 & 2 & | & 4\\ 0 & 1 & 2 & | & 3 \end{array}\right]\left\{\begin{array}{l} .\\ .\\ \leftarrow R_{3}-R_{2}. \end{array}\right.$ $\left[\begin{array}{lllll} 1 & 2 & 4 & | & 8\\ 0 & 1 & 2 & | & 4\\ 0 & 0 & 0 & | & -1 \end{array}\right]$ (inconsistent, the last row represents the impossible equation $0=-1)$ There are no a,b,c such that $w=av_{1}+bv_{2}+cv_{3}$, so $w\not\in$Span$S$