• عربي

Need Help?

Subscribe to Linear Algebra

###### \${selected_topic_name}
• Notes

Homogeneous system

$\left(\begin{array}{lll|l}{1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {0}\end{array}\right)$

$\left(\begin{array}{l}{x_{1}} \\ {x_{2}} \\ {x_{3}}\end{array}\right) =\left(\begin{array}{l}{0} \\ {0} \\ {0}\end{array}\right)$

trivial sol.

$\left(\begin{array}{lll|l}{1} & {a} & {b} & {0} \\ {0} & {1} & {c} & {0} \\ {0} & {0} & {1} & {0}\end{array}\right)$

$\left(\begin{array}{lll|l}{1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {0} & {0}\end{array}\right)$

$\left(\begin{array}{lll|l}{1} & {0} & {1} & {0} \\ {0} & {1} & {2} & {0} \\ {0} & {0} & {0} & {0}\end{array}\right)$

Let $x_{3}=s$

$x_{1}=-s$
$x_{2}=-2s$

$\left(\begin{array}{l}{x_{1}} \\ {x_{2}} \\ {x_{3}}\end{array}\right)=s\left(\begin{array}{c}{-1} \\ {-2} \\ {1}\end{array}\right)$

$\left(\begin{array}{lllll|l}{1} & {0} & {0} & {1} & {4} & {0} \\ {0} & {1} & {0} & {2} & {5} & {0} \\ {0} & {0} & {1} & {3} & {6} & {0}\end{array}\right)$

$x_{4}=r$
$x_{5}=t$

$x_{1}=-r-4t$
$x_{2}=-2 r-5 t$
$x_{3}=-3r-6 t$

$\left[\begin{array}{l}{x_{1}} \\ {x_{2}} \\ {x_{3}} \\ {x_{4}} \\ {x_{5}}\end{array}\right]=r\left[\begin{array}{c}{-1} \\ {-2} \\ {-3} \\ {1} \\ {0}\end{array}\right]+t\left[\begin{array}{c}{-4} \\ {-5} \\ {-6} \\ {0} \\ {1}\end{array}\right]$