I saw the sentence like this on the internet.
I went to the kitchen, cooked for me.
Can I understand the meaning of this sentence like this?
I went to the kitchen and I cooked for me.
Have I made a correct paraphrase?
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The quote looks like a form of colloquial speech, where the subject of the sentence (the first-person pronoun) is dropped.
Sentences in English are generally considered to require overt subjects. In a standard description of English, (1a) below would be considered grammatical, while (1b) would be considered ungrammatical:
(1) (a) I walked the dog yesterday.
(b) Walked the dog yesterday.
But in fact this is not true. (1b), and sentences like it lacking subject pronouns, are in fact common in colloquial spoken English.
With this in mind, a more traditionally correct version of your quote would be:
This agrees with your interpretation.
cooked for dinner.
Clearly this is not very good English.
What the writer did was make the reader "understand" too much.
The first "understanding" is that "I" cooked, even though the "I" is in the first phrase. So we have:
I cooked for dinner.
Now "understand" "for dinner" is what was done to have dinner, not what was eaten.
I went to the kitchen, (and) for dinner, I cooked.
I went to the kitchen, and in order to have dinner, I cooked.
I would not advise writing English in such "shorthand"(understandings).
You got it right, however, it is grammatically incorrect. On the other hand, such usage of "me" in place of "myself" is informal and I sometimes see in English:
Unless you wish to sound ungrammatical it is better to say: