Ég er þegar búinn með verkefnið.

Here þegar means already.

In this sentence, it shows that the assignment was finished before now, and that this is relevant at the present moment. Its position is also very natural: in a main clause, Icelandic often puts short adverbs like þegar after the finite verb, so Ég er þegar búinn ... is a normal word order.

Because vera búinn is a very common Icelandic way to talk about something that has been completed.

Literally, Ég er þegar búinn með verkefnið is closer to I am already finished with the assignment than to I have already finished the assignment. But in real usage, it often covers the same idea that English expresses with the present perfect.

So:

• Ég er þegar búinn með verkefnið = everyday, very natural
• Ég hef þegar lokið verkefninu = more formal/literary style

For learners, it is best to treat vera búinn as one of the most useful everyday patterns in Icelandic.

Búinn means something like finished, done, or completed, depending on context.

In this sentence, it describes the speaker’s state: the speaker is now in the state of having finished the assignment.

That is why Icelandic uses er + búinn:

• er = am/is/are
• búinn = finished/done

So the structure is very adjective-like.

Yes — the form changes depending on the gender and number of the subject, not the object.

Here, búinn is masculine singular, so it would be used by a male speaker.

Other forms include:

• Ég er búinn = a man says I am finished
• Ég er búin = a woman says I am finished
• Við erum búnir = a mixed group or all-male group says we are finished
• Við erum búnar = an all-female group says we are finished

So if a female speaker says this sentence, it would normally be:

Ég er þegar búin með verkefnið.

Not really in a literal English-style with sense. In this sentence, með is part of the fixed expression vera búinn með e-ð, which means to be finished with something.

So you should learn this as a chunk:

• vera búinn með + noun = be finished with + noun

Examples:

• Ég er búinn með kaffið. = I’m done with the coffee.
• Hann er búinn með bókina. = He has finished the book.

So með here is idiomatic, not a simple word-for-word with.

This is a very common question, and both patterns are important.

1. vera búinn með + noun

This focuses on the thing that is finished.

• Ég er búinn með verkefnið. = I’m finished with the assignment.

2. vera búinn að + infinitive

This focuses on the action that has been completed.

• Ég er búinn að klára verkefnið. = I have finished doing the assignment / I’ve finished the assignment.

Both can describe the same situation, but the structure is different:

• með + noun → finished with something
• að + infinitive → finished doing something

Verkefni means assignment, task, or project, depending on context.

The ending -ið is the definite article attached to the noun, so:

• verkefni = an assignment / a task
• verkefnið = the assignment / the task

Unlike English, Icelandic usually adds the directly onto the end of the noun instead of using a separate word.

So verkefnið is simply the definite form.

In this expression, it is best to learn the pattern as:

búinn með + accusative

So the noun after búinn með is normally treated like an object of the whole expression.

That can be confusing, because learners often know that ordinary prepositional með usually goes with the dative. But in this idiomatic construction, you should learn the phrase as a whole.

With verkefni, the definite singular form verkefnið does not visibly show the difference between nominative and accusative, so the case is not obvious just from this word alone.

A clearer example is:

• Ég er búinn með bókina.
  Here bókina is accusative.

So the safest learner rule is:

• vera búinn með + accusative

Because Icelandic main clauses normally follow the verb-second rule.

That means the finite verb usually comes early in the sentence, very often in the second position. Here:

• Ég = first element
• er = finite verb in second position
• þegar = adverb after the verb
• búinn með verkefnið = rest of the sentence

So Ég er þegar búinn ... is the natural structure.

For an English speaker, the position of þegar may feel slightly different, but it works much like already in I am already done.

Yes. A more formal alternative is:

Ég hef þegar lokið verkefninu.

This uses the verb ljúka meaning to complete / finish, and it sounds more formal or written than vera búinn.

So roughly:

• Ég er þegar búinn með verkefnið. = everyday, conversational
• Ég hef þegar lokið verkefninu. = more formal

Most learners will hear and use vera búinn very often in normal speech.

You would say:

Ég er ekki búinn með verkefnið enn.

If the speaker is female:

Ég er ekki búin með verkefnið enn.

Breakdown:

• ekki = not
• enn = yet / still

So:

• Ég er ekki búinn með verkefnið. = I’m not finished with the assignment.
• Ég er ekki búinn með verkefnið enn. = I’m not finished with the assignment yet.

Not always. Verkefni is a flexible word and can mean:

• assignment
• task
• project
• sometimes exercise or piece of work, depending on context

So in different situations, verkefnið could refer to schoolwork, a work project, or some task someone was supposed to complete.

That is very normal in Icelandic: one word often covers a range that English splits into several words.