Writing a vector as a linear combination of vectors from another basis

by Dominic Hicks   Last Updated October 17, 2018 00:20 AM

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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