[Math] Find a finite set of vectors which spans $W$.

Question

Let $W$ be the set of all $(x_1, x_2, x_3, x_4, x_5)$ in $\Bbb R^5$ which satisfy
$2x_1-x_2+{4 \over 3}x_3 – x_4\qquad = 0$,
$x_1\qquad+{2 \over 3}x_3\qquad- x_5 = 0$,
$9x_1-3x_2+6x_3-3x_4-3x_5 = 0$.
Find a finite set of vectors which spans $W$.
I have found the solution set which is $({-2\over3}c + e, 2e-d, c, d, e)$. Then will the required vectors which spans the solution space be $({-2\over3}, 0, 1, 0, 0), (0, -1, 0, 1, 0), (1, 2, 0, 0, 1)$? Is this the right way of solving the problem or is there a better way?

Answer 1

You can form a matrix and reduce it:

$$\begin{pmatrix} 1&0&\frac23&0&\!\!-1\\2&\!\!-1&\!\!-\frac43&\!\!-1&0\\ 9&\!\!-3&6&\!\!-3&\!\!-3\end{pmatrix}\longrightarrow\begin{pmatrix} 1&0&\frac23&0&\!\!-1\\ 0&\!\!-1&\!\!-\frac83&\!\!-1&2\\ 0&\!\!-3&0&\!\!-3&6\end{pmatrix}\longrightarrow\begin{pmatrix} 1&0&\frac23&0&\!\!-1\\ 0&\!\!-1&\!\!-\frac83&\!\!-1&2\\ 0&0&8&0&0\end{pmatrix}$$

And the solution space of the above is two dimensional, not three dimensional as what you got.