Q:

Tafsir of Verse 13 in Sura 3

In Sura 3, Verse 13, the verse in English translation is 

Already there has been for you a sign in the two armies which met -
  one fighting in the cause of Allah and another of disbelievers. They
  saw them [to be] twice their [own] number by [their] eyesight. But
  Allah supports with His victory whom He wills. Indeed in that is a
  lesson for those of vision.

I know that the context of this verse is Battle of Badr. But who saw who twice ? The Muslims or disbelievers ?

A:

The matter can be explained by other verses were Allah says:

And already had Allah given you victory at [the battle of] Badr while you were few in number. Then fear Allah ; perhaps you will be grateful. (3:123)

And [remember] when He showed them to you, when you met, as few in your eyes, and He made you [appear] as few in their eyes so that Allah might accomplish a matter already destined. And to Allah are [all] matters returned. (8:44)

These verses show that in fact the Muslims were few, but Allah showed both parties smaller to each other for reasons we are not to discuss here and which are the topic of this question: What is the secret behind showing both adversaries as few in the others eyes in the battle of Badr? (unanswered so far).  
Nevertheless both interpretations may come to mind if we recite verse (3:13) in different qira'at without having prior knowledge of the grammatical position of the verb and the observer or "acting person" as meant by the scholars of qir'aat.
But first we must be aware that the beginning of the verse:

Already there has been for you a sign in the two armies which met 

"you" here is referring to the disbelievers or the Jews (according ibn Kathir).  
Further note that the qira'at of this verse differ especially when it comes to the verb "to see" and its context, the majority recites:

يَرَوْنَهُمْ

which can be translated: 

they see them 

This is referring to the Muslims who see the army of the Kufar (smaller than it actually is or in the context of our discussed verse twice as many as they were themselves). So it describes the perspective of the Kufar (Muslims: "they"; Kufar: "you).
And Nafi', his teacher abu Ja'afar and Yaqoob recite:

تَرَوْنَهُمْ 

Which can be translated: 

you see them

referring to the Muslims, but changing the observer: here the Muslims are addressed and it is them who observe the Kufar (Muslims: "you"; Kufar: "they").
Ibn Kathir (for example) quoted in his tafsir the following interpretations:

They saw them with their own eyes twice their number)
  means, the idolators thought that the Muslims were twice as many as they were, for Allah made this illusion a factor in the victory that Islam had over them.
  It was said that the meaning of Allah's statement,
  (They saw them with their own eyes twice their number)
  is that the Muslims saw twice as many idolators as they were, yet Allah gave them victory over the disbelievers. 'Abdullah bin Mas'ud said, "When we looked at the disbelievers' forces, we found that they were twice as many as we were. When we looked at them again, we thought they did not have one man more than we had. (Source qtafsir)

