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.



Related Questions



Confusion on dimensions of basis

Updated March 29, 2017 03:20 AM

Compute the change of basis matrix in Matlab

Updated April 01, 2017 03:20 AM

Find a basis for a kernel

Updated April 01, 2017 08:20 AM

Determine the vectors of components

Updated December 10, 2016 08:08 AM