Not much is known about Lina's family.
There is only one known person in the anime series, and that's her older sister, Luna.
Lina is deathly afraid of Luna. When Luna once wrote a simple letter to Lina,
she built a giant pyramid and buried herself in it, signifying the magnitude of
fear her sister's presence evokes in her.Why is Lina so afraid of her older sister? What happened that triggered this sisterly fear? Well, let me explain. Life was tough for Lina and Luna's family while they were growing up, so Luna got a job as a waitress to support herself. Lina, always the clever businessman, thought she might help out by selling videos of Luna taking a bath. Luna wasn't pleased, to say the least. Lina ended up running away and is afraid of what Luna might do as revenge in return of that episode.
According to Filia, there's a rumor that Luna's much stronger than Lina. Strong, feared sorceresses just run through the genetics of the Inverse family, don't they? There are hints throughout the anime series that Luna was rough with Lina. For example, Lina was once electrocuted pretty badly by an opponent, and she said, "Compared to what my sister used to put me through, that was nothing!" There are other things hinting that Luna was harsh towards Lina. However, Lina once said that her sister is strict but always has a reason to be.
Luna is also the Sefield Knight. That proves, hands down, that she's extremely powerful. The Sefield Knight is a legend in itself. This fact could be another one of the reasons Lina is so afraid of Luna. Luna is actually more powerful than Lina herself! Now that's a scary thought. Imagine if the two battled off. Instant massive destruction would ensue. Luna using the magic of the Sefield Knight, Lina the Lord of Nightmares, oh my.