The set given by
$$S = \left\{ \begin{bmatrix} -d+2c+2b+a \\ 3d+3c+6b+3a \\ d+c+2b+a \\ -25d+5c-10b-5a \end{bmatrix} : a,b,c,d \in \mathbb{R} \right\}$$
is a subspace of $\mathbb{R}^4$. A basis for $S$ is given by
$$B = \left\{ \begin{bmatrix} \\ \\ \\ \end{bmatrix}, \begin{bmatrix} \\ \\ \\ \end{bmatrix}, \begin{bmatrix} \\ \\ \\ \end{bmatrix}, \begin{bmatrix} \\ \\ \\ \end{bmatrix} \right\}$$
