Ich winke meiner Mutter zurück.

Because winken normally takes the person you wave to in the dative case.

So:

• ich winke meiner Mutter = I wave to my mother
• not ich winke meine Mutter

Here, meiner is the dative form of meine because Mutter is feminine singular.

A useful pattern is:

• jemandem winken = to wave to someone

So you can think of it as:

• Ich winke meinem Freund.
• Sie winkt ihrer Schwester.

Zurück means back or in return.

So Ich winke meiner Mutter zurück means that I am waving back to my mother, not just waving to her for the first time.

Compare:

• Ich winke meiner Mutter. = I wave to my mother.
• Ich winke meiner Mutter zurück. = I wave back to my mother.

Because this is a separable verb: zurückwinken = to wave back.

In a normal main clause, separable verbs split apart:

• Ich winke meiner Mutter zurück.

The conjugated part goes in the usual verb position, and the separable prefix goes to the end.

In the infinitive, they stay together:

• zurückwinken

Examples:

• Ich rufe dich an. → anrufen
• Er kommt zurück. → zurückkommen
• Ich winke zurück. → zurückwinken

Because the subject is Ich, so the verb must be in the first person singular.

The present tense of winken is:

• ich winke
• du winkst
• er/sie/es winkt
• wir winken
• ihr winkt
• sie/Sie winken

So Ich winke is the correct form for I wave / I am waving.

It is the inflected form of the possessive word mein.

Base form:

• mein = my

But in German, words like mein change depending on gender, case, and number.

Since Mutter is:

• feminine
• singular
• dative

the form becomes:

• meiner Mutter

So meiner means my here, but in the form required by the grammar of the sentence.

Because all German nouns are capitalized.

So:

• Mutter = mother
• Hund = dog
• Haus = house

This is true even when the noun is not at the beginning of the sentence.

Not normally in this meaning.

The standard German pattern is:

• jemandem winken

So the natural sentence is:

• Ich winke meiner Mutter.

Using zu would sound unnatural here for most learners’ intended meaning.

English says wave to someone, but German usually does not copy that structure directly. This is a common place where English speakers need to adjust.

Yes, depending on context, winken can be used for both kinds of hand gesture.

But very often:

• jemandem winken = to wave to someone
• jemanden heranwinken / zu sich winken = to beckon someone over

In your sentence, with zurück, the meaning is clearly wave back.

Yes, some parts can move, but the separable prefix zurück still stays at the end in a main clause.

For example:

• Ich winke meiner Mutter zurück.
• Meiner Mutter winke ich zurück.

The second version puts extra emphasis on meiner Mutter.

But you would not normally say:

• Ich zurück winke meiner Mutter.

That is incorrect in a standard main clause.

The full dictionary form is zurückwinken, because the sentence means to wave back.

In the sentence, it appears split:

• winke ... zurück

If you were looking it up, both of these are useful:

• winken = to wave
• zurückwinken = to wave back

So the sentence uses the separable verb zurückwinken in the first person singular present.