TITLE: Octagon symmetry and base change
QUESTION [1 upvotes]: I'm having trouble with this question in my course.
I am to consider a regular octagon with vertices A, B, C, D, E, F, G and H in counter clockwise order. 
The vector $\overrightarrow{u}$, wich has the coordinates $(1,2)$ in the base $e_1=\overrightarrow{AB}$ and $e_2=\overrightarrow{AE}$. I am supposed to express $\overrightarrow{u}$ coordinates in the base $f_1=\overrightarrow{AC}$, $f_2=\overrightarrow{AH}$.
I have done this exercise in a regular hexagon. But I do not know how to derive the symmetries in an octagon to so that I can express the vector in the new base.
Need some good guidance,
Many thanks.

REPLY [0 votes]: (Hint) Let's say you have an octagon like this:

Consider a coordinate system such that $AB=\sqrt2$ lies along the $x$ axis and $A$ is the origin. In the standard basis, $e_1=(\sqrt2,0)$, $e_2=(\sqrt2,2+\sqrt2)$.
You have $u=e_1+2e_2$, which will give you the components of $u$. Similarly, can you find the standard basis components of $f_1$ and $f_2$ as we did for $e_1$ and $e_2$? After that, assuming $u=af_1+bf_2$, you can solve for $a$ and $b$ via a pair of simultaneous equations.
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