# [SOLVED]311.1.5.5 homogeneous systems in parametric vector form.

#### karush

##### Well-known member
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

#### Country Boy

##### Well-known member
MHB Math Helper
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

##### Well-known member
ok i tried to follow a hand written example in saw on Google images