Find a Basis for the Set of All Solutions to the Difference

Multiple Choice

Find a basis for the set of all solutions to the difference equation.
-Consider two bases $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 } \right\}$ and $C = \left\{ \mathbf { c } _ { 1 } , \mathbf { c } _ { 2 } , \mathbf { c } _ { 3 } \right\}$ for a vector space $V$ such that $\mathbf { b } _ { 1 } = \mathbf { c } _ { 1 } + 2 \mathbf { c } _ { 3 } , \mathbf { b } _ { 2 } = \mathbf { c } _ { 1 } + 5 \mathbf { c } _ { 2 } - \mathbf { c } _ { 3 }$ , and $\mathbf { b } _ { 3 } = 3 \mathbf { c } _ { 1 } - \mathbf { c } _ { 2 }$ . Suppose $\mathbf { x } = \mathbf { b } _ { 1 } + 4 \mathbf { b } _ { 2 } + \mathbf { b } _ { 3 }$ . That is, suppose $[ \mathbf { x } ] _ { B } = \left[ \begin{array} { l } 1 \\ 4 \\ 1 \end{array} \right]$ . Find $[ \mathbf { x } ] _ { C }$ .

A)
$\left[ \begin{array} { r } 8 \\ 19 \\ - 2 \end{array} \right]$

B)
$\left[ \begin{array} { r } 3 \\ 20 \\ - 1 \end{array} \right]$

C)
$\left[ \begin{array} { r } 8 \\ 19 \\ 6 \end{array} \right]$

D)
$\left[ \begin{array} { r } 8 \\ 21 \\ - 4 \end{array} \right]$