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$\tiny{311.1.7.9}$

For what values of $h$ is $v_3$ in Span $\{v_1,v_2,v_3\}$ linearly \textit{dependent}. Justify

$v_1=\left[\begin{array}{rrrrrr}1\\-3\\2\end{array}\right],

v_2=\left[\begin{array}{rrrrrr}-3\\9\\-6\end{array}\right],

v_3=\left[\begin{array}{rrrrrr}5\\-7\\h\end{array}\right]$

$v_3$ is in Span{v1, v2} means there exists a constant such that

$c_1v_1 + c_2v_2 = v_3$

So if, but this is an augmented matrix

$\left[\begin{array}{rr|r}1 &−3& 2 \\ −3 &9 &−7 \\5 &−7& h \end{array}\right]$

RREF

$\left[ \begin{array}{ccc} 1 & -3 & 2 \\0 & 0 & -1 \\0 & 8 & h - 10 \end{array} \right]$

anyway, so far:unsure:$

For what values of $h$ is $v_3$ in Span $\{v_1,v_2,v_3\}$ linearly \textit{dependent}. Justify

$v_1=\left[\begin{array}{rrrrrr}1\\-3\\2\end{array}\right],

v_2=\left[\begin{array}{rrrrrr}-3\\9\\-6\end{array}\right],

v_3=\left[\begin{array}{rrrrrr}5\\-7\\h\end{array}\right]$

$v_3$ is in Span{v1, v2} means there exists a constant such that

$c_1v_1 + c_2v_2 = v_3$

So if, but this is an augmented matrix

$\left[\begin{array}{rr|r}1 &−3& 2 \\ −3 &9 &−7 \\5 &−7& h \end{array}\right]$

RREF

$\left[ \begin{array}{ccc} 1 & -3 & 2 \\0 & 0 & -1 \\0 & 8 & h - 10 \end{array} \right]$

anyway, so far:unsure:$

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