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I have the bases $B=\{\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 2 \end{pmatrix}\}$ and $C=\{\begin{pmatrix} -4 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \end{pmatrix}\}$.

I'm asked to write the vector $3\begin{pmatrix} 1 \\ 1 \end{pmatrix} - 2\begin{pmatrix} -1 \\ 2 \end{pmatrix}$ as a linear combination of the vectors from the basis $C$.

I don't understand how this is even possible. Using just the two vectors from $C$ I can't seem to get the result $\begin{pmatrix} 5 \\ -1 \end{pmatrix}$ as needed. Is there something I'm missing? I've already found $P_{B\leftarrow C}$ and $P_{C\leftarrow B}$ but I'm not sure my answers are correct, and I'm not sure if the change of base matrices are even relevant here to express this linear combination.

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Use a system of linear equations. We see that we want to solve the system $x_1$(-4,2) + $x_2$(2,5) = (5,-1). I don't think there is any need to discuss similar matrices.

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