"You know that I love you."
Translation:Du vet att jag älskar dig.
Depending on the context you could for example say that to your wife meaning that she knows you love her and your kids, or any other situation where you are talking to one person and saying you love a larger group they are a part of. I do agree that it may be a bit confusing in an exercise like this though.
With an out-of-context sentence like this it could also translate to "Ni vet att jag älskar er", as in addressing a group and loving the group. "Ni vet att jag älskar er" could also be addressing a 'formal you' and loving that person - a bit odd though being formal and intimate in the same sentence... :)
Back in the days Swedes used to be very polite and would address strangers or people "above" themselves differently than people they knew well, some still do. The pronoun "Ni" is then used instead of "du". It works the same way as "Sie" and "du" in German, and "I" and "du" in Danish I think.
I agree. 'Du vet att jag älskar er' is, maybe not ungrammatical, but at least inconsistent (unless 'du' = 1 person and 'er' = several). Ni/er, which is normally you (plural), may sometimes be you (singular formal), but i don't think that is very common usage. (Likewise, English historically had 'thou' (cognate to Swedish 'du') for you (sing.) and 'you' for you (pl.); 'you' came to be used for you (sing. formal) and eventually displaced 'thou'.) However, in an expression like 'You know that i love you' or its translation in other languages, both 'you's would in all likelihood refer to the same single person (or, less likely, both refer to the same group of multiple people). So, it should be 'Du vet att jag älskar dig'; dig = object form of du.
I don't know the grammatical terms 100%, but if you think about it in english, it's a different kind of "that".
In this sentence ("You know that I love you") the "that" is a conjuction. You have to say "that" in this sentence, you can't say "You know this I love you" or anything else.
But if "that" is used as pronoun and is interchangeable with "this" or "it", then you use "det". Example: "Jag vet det" - "I know this/that/it". That's how I understand it. I hope that is correct and makes sense?
Comparing it to German, I feel like att=dass, and det=das.