# Finding coordinate vector relative to different bases

I have been working on coordinate vectors, change of basis, and matrix of linear transformation. However, I don't know how to find the coordinate vector with respect to different bases. Any help would be greatly appreciated!

Problem statement: Let $B=(e_1+e_2,e_1-e_2), B'=(-2e_2,e_1+e_2)$. Both $B, B'$ are bases for $R^2$. Suppose $v \in R^2$ and the coordinate vector of $v$ with respect to $B$ is the column vector [-2 6]^t. I have to find the coordinate vector of $v$ with respect to $B'$.

Thanks!

## 1 Answer

\begin{aligned} v&=-2(e_1+e_2)+6(e_1-e_2)\\ &=-2e_1-2e_2+6e_1-6e_2\\ &=4e_1-8e_2\\ &=6(-2e_2)+4(e_1+e_2) \end{aligned} hence new coordinates are (6,4).