Understanding basis algorithm result I've a matrix ${\bf A}$ defined as
A = \begin{pmatrix}
        1 & -2 & 0 & 3 & 7\\
        2 & 1 & -3 & 1 & 1\\
        \end{pmatrix}
And ${W_1}$ is the solution space associated to the homogenous system ${\bf A}$. I am asked to find a basis for ${W_1}$.
I re-wrote the matrix as linear combination of ${c_1}$, i.e.:
\begin{align}
x = c_1 + 2 c_2 \\
y = -2c_1+ c_2 \\
z = -3c_2 \\
v = 3c_1 + c_2 \\
w = 7c_1 + c_2 \\
\end{align}
I use these equations to build another matrix, I solve it and then i reach to:
\begin{align}
x - \frac{2}{3}y = c_1 \\
x - \frac{-1}{3}z = c_2 \\
2x + y \frac{-5}{3}z = 0 \\
x - y - v = 0 \\
3x - 2y - 2 = 0
\end{align}
So, the last three equations are the general ones for $W_1$ and from there I can describe the basis, is this right ?

A standard Gaussian elimination leads to a reduced row echelon form:
\begin{align}
\begin{pmatrix}
1 & -2 & 0 & 3 & 7\\
2 & 1 & -3 & 1 & 1
\end{pmatrix}
&\to
\begin{pmatrix}
1 & -2 & 0 & 3 & 7\\
0 & 5 & -3 & -5 & -13
\end{pmatrix}
&&R_2\gets R_2-2R_1
\\
&\to
\begin{pmatrix}
1 & -2 & 0 & 3 & 7\\
0 & 1 & -3/5 & -1 & -13/5
\end{pmatrix}
&&R_2\gets \frac{1}{5}R_2
\\
&\to
\begin{pmatrix}
1 & 0 & -6/5 & 1 & 9/5\\
0 & 1 & -3/5 & -1 & -13/5
\end{pmatrix}
&&R_2\gets \frac{1}{5}R_2
\end{align}
This means the system is equivalent to
$$
\begin{cases}
x_1=\dfrac{6}{5}x_3-x_4-\dfrac{9}{5}x_5 \\[6px]
x_2=\dfrac{3}{5}x_3+x_4+\dfrac{13}{5}x_5
\end{cases}
$$
Now it's just a matter of choosing values for $x_3$, $x_4$ and $x_5$ to obtain linearly independent vectors; the choices, in order to also remove denominators, can be
$$
x_3=5,\quad x_4=0,\quad x_5=0\\
x_3=0,\quad x_4=1,\quad x_5=0\\
x_3=0,\quad x_4=0,\quad x_5=5
$$
leading to the three vectors
$$
\begin{pmatrix}6\\3\\5\\0\\0\end{pmatrix},\quad
\begin{pmatrix}-1\\1\\0\\1\\0\end{pmatrix},\quad
\begin{pmatrix}-9\\13\\0\\0\\5\end{pmatrix}.
$$
These three vectors form a basis for $W_1$.
