How to find a solution for a matrix with 1 equation and 3 unknown variables? The task is to find all solutions for $A_1 x = 0$ with $x\in \mathbb R^3$
$$A_1 = \begin{pmatrix}
6 & 3 & -9 \\
2 & 1 & -3 \\
-4 & -2 & 6 \\
\end{pmatrix}
$$
The given solution is as follows:
$$L_1 = \{ \lambda \begin{pmatrix}
1 \\
1 \\
1 \\
\end{pmatrix} + \mu \begin{pmatrix} 0 \\ 3 \\ 1 \\ \end{pmatrix} | \lambda, \mu \in \mathbb R \}$$
As far as I understand the matrix has 3 unkowns and only 1 equation resulting in: 
$$A_1 = \begin{pmatrix}
2 & 1 & -3 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}
$$
My approach was to find a way to display the soltions by rearranging $x_1, x_2$ and $x_3$ in dependancy to each other, which results in:
$$L_1 = \{x =  \begin{pmatrix} \frac{3x_3-x_2}{2} \\ 3x_3-2x_1 \\ \frac{2x_1 + x_2}{3} \end{pmatrix} \space | \space x \in \mathbb R^3\}$$
I tried searching for a similar problem but usually there are at least two equations for three unknowns resulting in one free to choose and the others depending on the one chosen.
 A: I don't understand how you can represent the solutions of the linear equation 
$$2x_1+x_2-3x_3=0$$ 
as
$$L_1 = \left\{x =  \begin{pmatrix} \frac{3x_3-x_2}{2} \\ 3x_3-2x_1 \\ \frac{2x_1 + x_2}{3} \end{pmatrix} \space | \space x \in \mathbb R^3\right\}.$$
You may write for example,
$$L_1 = \left\{\begin{pmatrix} \frac{3x_3-x_2}{2} \\ x_2 \\ x_3 \end{pmatrix} \space | x_2, x_3 \in \mathbb{R}\right\}
=\{ x_2 \mathbf{u} + x_3 \mathbf{v} | x_2, x_3 \in \mathbb{R} \}$$
where 
$$ \mathbf{u}=(-1/2,1,0)^t\quad\mathbf{v}=(3/2,0,1)^t.$$
Similarly
$$L_1 = \left\{\begin{pmatrix} x_1 \\ 3x_3-2x_1 \\ x_3\end{pmatrix} \space | \space x \in \mathbb R^3\right\}
=\{ x_1 \mathbf{u} + x_3 \mathbf{v} | x_1, x_3 \in \mathbb{R} \}$$
where 
$$ \mathbf{u}=(1,-2,0)^t\quad\mathbf{v}=(0,3,1)^t.$$
The set of all solutions of the given equation is a vector space of dimension $2$ in $\mathbb{R}^3$. We find two linearly independent vectors $\mathbf{u}$, and $\mathbf{v}$ which satisfy the equation and the set of solutions can be written as
$$L_1 = \{ \lambda \mathbf{u} + \mu \mathbf{v} | \lambda, \mu \in \mathbb{R} \}.$$
