I am not sure when ではありません and 〜くない are used. I was given the sentence この赤いかばんは大きいではありません (as in "this red bag is not big") and I am not sure why この赤いかばんは大きくないです was not used. Could anyone explain? Thanks!
Are you sure this is the sentence you saw and that it came from a reputable source?
The sentence does not mean "This red bag is not big". As you correctly pointed out, the correct way to say that would be この赤いかばんは大きくありません or この赤いかばんは大きくないです etc.
You cannot negate an i-adjective by adding ではありません, ではない、じゃない etc, after the adjective.
This sentence has the opposite meaning of "This red bag is big, isn't it?", but expressed rather formally. This link may be helpful.