How to express transmitted EM wave amplitude from polarization basis

[happyparticle]

Hi,
I have an expression in my textbook that I don't really understand.
I have 2 questions regarding this expression for a linear EM wave$\tilde{\vec{E_{0i}}} = (E_{0x} \hat{x} \pm E_{0y} \hat{y}) e^{i(kz- \omega t)}$
$\tilde{\vec{E_{0t}}} = (\sum_j E_{oij} e_{pj}) \hat{e_p}$
$\tilde{\vec{E_{0t}}} = E_0 cos \theta \hat{x} + E_0 sin \theta \cdot 0$ where $\hat{p_j} = \hat{x}$

First of all, is it the total electric wave transmitted? and secondly, can someone explain me how we get the last expression from de second?

 

[Charles Link]

Can you give a source where this came from? It looks like class notes where they may have made a mistake and/or left out a step or two.

 

[DaveE]

Yes, what @Charles Link said. Those subscripts are really confusing. We need to see more.