PCA loadings interpretation

Question

Regarding the final output of PCA:

1. Using rotation, I'll get a loading for my 50 components, so the equation would be $pc_1=0.05*a_1+0.02*a_2+\dots+0.04*a_{50}$. Now, can I derive the value of $pc_1$ by using my last observed return for $a_1,a_2,\dots,a_{50}$?

2. As I looked at my data, I have found at least 10 principal components are required to explain 80% of the variance, so using same logic as above, I'll get $pc_1,pc_2,\dots,pc_{10}$. Now the regression would be $y=w_1*pc_1+w_2*pc_2+...+w_{10}*pc_{10}+e $. In each of them $w_1,w_2,\dots,w_{10}$ can be their respective eigenvalue and the remaining term is an error term?

Answer 1

Everything sounds pretty OK except the ending of Q2. When you do regress y on the first ten PCs, w1,...,w10 are unknown weights to be estimated from your data using Ordinary Least Squares for example.