Kennarinn biður mig að athuga heimildina vandlega.

Kennari means teacher in the indefinite sense, while kennarinn means the teacher.

Icelandic usually puts the definite article on the end of the noun instead of using a separate word like English the:

• kennari = teacher
• kennarinn = the teacher

So Kennarinn biður mig... means The teacher asks me...

The infinitive is biðja = to ask / to request / to beg.

In this sentence, kennarinn is a third-person singular subject (the teacher), so the verb has to match that:

• ég bið = I ask
• þú biður = you ask
• hann / hún / það biður = he / she / it asks

So:

• Kennarinn biður = The teacher asks

This verb is a little irregular, so learners often wonder why it is not something more predictable.

Because mig is the accusative form of ég (I).

After biðja in this pattern, the person being asked goes in the accusative:

• biðja einhvern að gera eitthvað
  = to ask someone to do something

So:

• mig = me (accusative)
• Kennarinn biður mig... = The teacher asks me...

Not:

• ég = I (subject form)
• mér = me (dative)

A very useful pattern is:

biðja einhvern að + infinitive

That means:

to ask someone to do something

So this sentence breaks down like this:

• Kennarinn = the teacher
• biður mig = asks me
• að athuga heimildina = to check the source
• vandlega = carefully

This is one of the most common ways to express ask someone to do something in Icelandic.

Here, að is the infinitive marker, so it works like English to before a verb:

• að athuga = to check

It does not mean the preposition at / by / to here, and it does not mean that either. Its job in this sentence is simply to introduce the infinitive verb.

So:

• biður mig að athuga = asks me to check

Because after að in this construction, Icelandic uses the infinitive.

• athuga = to check / to examine / to look into

If it were a finite verb with its own subject, then you would see a personal ending such as athugar. But here it is part of the phrase að athuga = to check.

So:

• Kennarinn biður mig að athuga...
  = The teacher asks me to check...

Heimildina is the definite accusative singular form of heimild.

Base form:

• heimild = source, reference, authority, permission (depending on context)

In this sentence it is the direct object of athuga, so it appears in the accusative, and it is definite:

• heimild = a source
• heimildina = the source

So:

• að athuga heimildina = to check the source

In this sentence, heimild most naturally means source or reference, especially in an academic or informational context.

But this word can have several related meanings depending on context, such as:

• source
• reference
• authority
• authorization / permission

Here, since someone is being asked to check it, source is the most likely meaning.

Vandlega is an adverb, and it means carefully.

It comes from the adjective vandlegur = careful / thorough. A lot of Icelandic adverbs are formed with -lega, similar to English -ly:

• vandlegur = careful
• vandlega = carefully

So:

• athuga heimildina vandlega = check the source carefully

That is a very natural position for an adverb in Icelandic. It is modifying the action að athuga heimildina.

So the sentence flows as:

• asks me
• to check the source
• carefully

Putting vandlega at the end is completely normal and idiomatic here.

Yes, very much so. The structure is quite close to English:

• Kennarinn = The teacher
• biður mig = asks me
• að athuga = to check
• heimildina = the source
• vandlega = carefully

So the pattern is essentially:

subject + verb + object + infinitive phrase + adverb

That makes this a very useful sentence for learners, because although Icelandic has case endings and different verb forms, the overall structure is fairly easy for an English speaker to follow.