= Solution
The <pivot columns> of the <reduced row echelon form> are columns 1 and 3. Take those columns of the original <matrix>, since <elementary row operations> need not preserve the original <column space>:
$$
\boxed{\left\{\begin{pmatrix}-1\\1\\2\end{pmatrix},\begin{pmatrix}3\\2\\5\end{pmatrix}\right\}.}
$$
For an explicit spanning check, column 2 is $-2a_1$, column 4 is $-a_1+2a_3$, and column 5 is $3a_1-2a_3$. The first two coordinates of the displayed columns form a $2\times2$ <matrix> of <determinant> $-5\ne0$, proving their <linear independence>. They are therefore a <basis> of the <column space>.
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