311.1.5.17 geometric description

[karush]

$\tiny{311.1.5.17}$
Give a geometric description of the solution set.

$\begin{array}{rrrrr}
-2x_1&+2x_2&+4x_3&=0\\
-4x_1&-4x_2&-8x_3&=0\\
&-3x_2&-3x_3&=0
\end{array}$
this can be written as
$\left[\begin{array}{rrr|rr}-2&2&4&0\\-4&-4&-8&0\\&-3&-3&0\end{array}\right]$

$\text{RREF}=\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right]$
ok I am not sure how you get a geometric description with everything going to zero

 

[skeeter]

the point (0,0,0)

 

[HOI]

Equivalently the matrix \(\displaystyle \begin{bmatrix}-2 & 2 & 4 \\ -4 & -4 & 8 \\ 0 & -3 & -3\end{bmatrix}\) has determinant \(\displaystyle \left|\begin{array}{ccc}-2 & 2 & 4 \\ -4 & -4 & 8 \\ 0 & -3 & -3 \end{array}\right|= 3\left|\begin{array}{cc}-2 & 4 \\ -4 & 8\end{array}\right|- 3\left|\begin{array}{cc}-2 & 2 \\ -4 & -4\end{array}\right|= 3(-16+ 16)- 3(8+ 8)= -3(16)= -48 \) which is not 0 so is a "one to one" transformation. The only vector that is mapped to the 0 vector is the 0 vector itself.