311.1.5.5 homogeneous systems in parametric vector form.

[karush]

Write the solution set of the given homogeneous systems in parametric vector form.
$\begin{array}{rrrr}
-2x_1& +2x_2& +4x_3& =0\\
-4x_1& -4x_2& -8x_3& =0\\
&-3x_2& -3x_3& =0
\end{array}\implies
\left[\begin{array}{rrrr}
x_1\\x_2\\x_3
\end{array}\right]
=\left[\begin{array}{rrrr}
-2\\-4\\\color{red}{0}
\end{array}\right]x_1
+\left[\begin{array}{rrrr}
2\\-4\\-3
\end{array}\right]x_2
+\left[\begin{array}{rrrr}
4\\-8\\-3
\end{array}\right]x_3$
red is a null space

ok its looks straight forward but still ? typos etc
is there an online calculator to check these
no book answer on this one

 

[HOI]

Write the solution set of the given homogeneous systems in parametric vector form.
$\begin{array}{rrrr}
-2x_1& +2x_2& +4x_3& =0\\
-4x_1& -4x_2& -8x_3& =0\\
&-3x_2& -3x_3& =0
\end{array}\implies
\left[\begin{array}{rrrr}
x_1\\x_2\\x_3
\end{array}\right]
=\left[\begin{array}{rrrr}
-2\\-4\\\color{red}{0}
\end{array}\right]x_1
+\left[\begin{array}{rrrr}
2\\-4\\-3
\end{array}\right]x_2
+\left[\begin{array}{rrrr}
4\\-8\\-3
\end{array}\right]x_3$
red is a null space

ok its looks straight forward but still ? typos etc
is there an online calculator to check these
no book answer on this one

No. The sum is not equal to "$\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}$. It is equal to "$\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$".
I also do not understand why you have written the "0" in red and called it a "null space". It is simply the number 0.

This is $\begin{bmatrix} -2 \\ -4 \\ 0 \end{bmatrix} x_1+ \begin{bmatrix} 2 \\ -4 \\ 3 \end{bmatrix} x_2+ \begin{bmatrix} 4 \\ -8 \\ -3 \end{bmatrix}x_3= \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}$.

 

[karush]

ok i tried to follow a hand written example in saw on Google images