(1) This is the house which she lives in. (a relative pronoun)
(2) This is the house (that) she lives in. (a relative pronoun)
(3) This is the house where she lives. (a relative adverb)
(4) This is the house she lives in. (omitting an objective relative pronoun)
(5) This is the house she lives. (omitting a relative adverb where)
(6) This is the house that she lives. (that is a kind of relative adverb where)
In a school grammar, the sentences from (1) to (4) are correct, but is it possible that the sentence from (5) or (6) is admitted as a correct sentence? If it is admitted in some places, to what extent is it acceptable? Can we teach that (5) and (6) sentences are acceptable and correct?
No: In Standard English (5) and (6) are ungrammatical. Note that strictly speaking, "that" is not a relative word but a subordinator.
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No: In Standard English (5) and (6) are ungrammatical.
Note that strictly speaking, "that" is not a relative word but a subordinator.
iwinners6) This is the house that she lives. (that is a kind of relative adverb where)
I think that the relative adverb "that" works when marking the time. For example, the year that she was born.
(I'm a non-native.)