Solve the Problem $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\}$

Multiple Choice

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

A)
$\left[ \begin{array} { l } - 5 \\ 17 \end{array} \right]$

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

C)
$\left[ \begin{array} { l } 15 \\ 10 \end{array} \right]$

D)

$\left[ \begin{array} { l } 16 \\ 10 \end{array} \right]$