From the text "Mathematical tools for applied multivariate analysis", page 113, there's an example of rotating axis and vectors. I don't understand how they've got the results from projecting the $e_i$ (standard basis vectors) onto the $f_i$ vectors (rotation of $e_i$).
Here's the diagram of the rotation:
Which shows we have $f_1' = (0.867 , 0.5)$, and $f_2' = ( -0.5, 0.867)$
Later in the text they state the following:
I don't see how this result is found though, if I try and project e1 -> f1 then I have:
e1 = (1, 0)
f1 = (0.867, 0.5)
theta = angle between f1 and e1
proj(e1 -> f1) = [ ( e1 . f1 ) / |f1|^2 ] . f1
= [ ( e1 . f1 ) / 1 ] . f1
= [ ( 0.867 ) ] . (0.867 0.5)
= (0.867^2 , 0.867 * 0.5)
= (.751, .43)
I'm clearly missing something obvious when projecting e1 onto f1, but I'm not sure what.
I can see that we'd move f1 by -0.5 units along the f2 axis to have it line up with e1, and that's where the numbers for $g_i$ come from, I just don't see how they've done the computation for them here.
First, I'm not quite sure why you wrote $f_1'$ and $f_2'$ with the primes.
Second, you're misinterpreting what $g_1$ and $g_2$ are. When we write $e_1=af_1+bf_2$, then $g_1=\left[\begin{matrix} a \\ b\end{matrix}\right]$, and similarly for $g_2$. Of course, $a=e_1\cdot f_1$ and $b=e_1\cdot f_2$.
Last, let me say that this is the most cluttered diagram I've seen in anything resembling a presentation of basic linear algebra.