chinguga naeil doseogwaneseo gati gongbuhajago haesseoyo.

Both are possible, but they give slightly different emphasis.

• 친구가 marks friend as the subject of the sentence: My friend said it.
• 친구는 would sound more like as for my friend... and may contrast with someone else.

In this sentence, 친구가 is the most natural neutral choice because you are simply stating who said it.

Because 공부하다 is an action, and 에서 marks the place where an action happens.

• 도서관에서 공부하다 = study at/in the library
• 도서관에 is more about location or destination:
  • 도서관에 가다 = go to the library
  • 도서관에 있다 = be at the library

So with study, 에서 is correct.

같이 means together.

So 도서관에서 같이 공부하자고 했어요 means your friend suggested that you study together at the library.

A very similar word is 함께, which also means together.
In everyday speech, 같이 is very common and natural.

공부하자 comes from 공부하다 and uses the ending -자, which means let’s ...

So:

• 공부하다 = to study
• 공부하자 = let’s study

This is a casual/proposal form used when suggesting that both speaker and listener do something together.

This is quoted speech.

• 공부하자 = let’s study
• -고 하다 = to say that...
• 공부하자고 했어요 = said let’s study / suggested studying

So the speaker is reporting what the friend said.

A useful pattern is:

• Verb-자고 하다 = to say let’s do..., to suggest doing...

Examples:

• 가자고 했어요 = said let’s go
• 먹자고 했어요 = said let’s eat

Here, -고 is part of the quotation pattern.

It connects the quoted content to 하다:

• 공부하자
  • 고 하다
• literally: say let’s study

So this -고 is not the same as the -고 that means and in sentences like 먹고 자요 (eat and sleep).
In this sentence, it is specifically a quoting marker.

Because the saying happened in the past, not the studying.

The timeline is:

• 내일 = tomorrow → the planned studying time
• 했어요 = said → the friend already said it

So the sentence means that earlier, your friend said/suggested that you study together tomorrow.

Korean often leaves out information that is obvious from context.

In English, you might say:

• My friend told me...
• My friend suggested to me...

In Korean, if it is clear who heard the suggestion, that part is often omitted.

If you wanted to include it, you could say something like:

• 친구가 저한테 내일 도서관에서 같이 공부하자고 했어요. = My friend said to me, let’s study together at the library tomorrow.

But leaving it out is very natural.

Korean usually puts time, place, and manner information before the main verb.

A common order is:

• time
• place
• manner
• verb

So here:

• 내일 = tomorrow
• 도서관에서 = at the library
• 같이 = together
• 공부하자고 했어요 = said let’s study

This is very normal Korean word order.

Yes. The sentence ends with 했어요, which is the polite informal style.

So the speaker is speaking politely to the listener.

However, inside the quotation, the friend’s original words are 공부하자, which is a casual suggestion form. That is normal, because quoted speech keeps the original style or a version close to it.

So the overall sentence is polite, even though the quoted suggestion itself is casual.

Yes. 같이 and 함께 both mean together.

• 같이 is very common in everyday conversation.
• 함께 can sound a little more formal or written, though it is also common in speech.

So both are correct:

• 친구가 내일 도서관에서 같이 공부하자고 했어요.
• 친구가 내일 도서관에서 함께 공부하자고 했어요.

The first one is probably what learners will hear more often in casual conversation.

This is an important difference.

• 공부하자고 했어요 = said let’s study
  → a suggestion including both speaker and listener
• 공부하라고 했어요 = said study / told someone to study
  → a command or instruction directed at the listener

So:

• -자고 하다 = suggest doing something together
• -라고 하다 = tell/order someone to do something

In your sentence, 공부하자고 했어요 shows a friendly proposal, not a command.