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

question 27

Multiple Choice

Find a basis for the set of all solutions to the difference equation.
-Let $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\}$ and $C = \left\{ \mathbf { c } _ { 1 } , \mathbf { c } _ { 2 } \right\}$ be bases for $\mathcal { R } ^ { 2 }$ , where
$$\mathbf { b } _ { 1 } = \left[ \begin{array} { l } 2 \\4\end{array} \right] , \mathbf { b } _ { 2 } = \left[ \begin{array} { l } 1 \\3\end{array} \right] , \mathbf { c } _ { 1 } = \left[ \begin{array} { l } 1 \\3\end{array} \right] , \mathbf { c } _ { 2 } = \left[ \begin{array} { r } - 4 \\- 10\end{array} \right]$$
Find the change-of-coordinates matrix from $B$ to $C$ .

• A)
• \(\left[ \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right]\)
• B)
• \(\left[ \begin{array} { l l } 0 & - 1 \\ 1 & - 2 \end{array} \right]\)
• C)
• \(\left[ \begin{array} { r r } 1 & - 4 \\ 3 & - 10 \end{array} \right]\)
• D)
• \(\left[ \begin{array} { l l } - 2 & 1 \\ - 1 & 0 \end{array} \right]\)

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