I am given the matrix $A=\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$,
and a vector $x=\begin{bmatrix} 1 \\ 1 \end{bmatrix}.$
After computing the eigenvalues and normalized eigenvectors, we can construct the orthonormal basis B: $B=\begin{bmatrix} \frac{1}{\sqrt{5}} & -\frac{2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix}$.
How can I compute the components of x in the orthonormal basis B? Should I use $x_{[B]} = B\cdot x$, or rather $x_{[B]} = B^{-1}\cdot x?$
I am a bit confused as to how we translate back and forth in different bases.