Find the dimension of and a basis for the subspace spanned by the following sets of vectors. Hint: First identify the subspace with the image of a certain matrix.
(a)
$$
\left(\begin{array}{r}
1 \\
2 \\
-1
\end{array}\right),\left(\begin{array}{l}
2 \\
2 \\
0
\end{array}\right) \text {, }
$$
(b)
$$
\left(\begin{array}{r}
1 \\
1 \\
-1
\end{array}\right),\left(\begin{array}{r}
2 \\
2 \\
-2
\end{array}\right),\left(\begin{array}{r}
-3 \\
-3 \\
3
\end{array}\right)
$$
(c) $\left(\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 2 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}1 \\ 2 \\ 3 \\ -3\end{array}\right)$,
(d) $\left(\begin{array}{r}1 \\ 0 \\ -3 \\ 2\end{array}\right),\left(\begin{array}{r}0 \\ 1 \\ 2 \\ -3\end{array}\right),\left(\begin{array}{r}-3 \\ -4 \\ 1 \\ 6\end{array}\right),\left(\begin{array}{r}1 \\ -3 \\ -8 \\ 7\end{array}\right),\left(\begin{array}{r}2 \\ 1 \\ -6 \\ 9\end{array}\right),(e)\left(\begin{array}{r}1 \\ 1 \\ -1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 2 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}3 \\ 0 \\ 1 \\ 3 \\ 2\end{array}\right),\left(\begin{array}{r}0 \\ -3 \\ 4 \\ 0 \\ -1\end{array}\right),\left(\begin{array}{r}1 \\ 3 \\ -1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 3 \\ 2 \\ 0\end{array}\right)$.