Learn Italian: Mia nonna è anziana ma vive da sola. - My grandmother is elderly but lives on her own.

While you are currently learning how da means "for" or "since" with time, it has many other jobs! Here, da solo or da sola is a fixed phrase that translates to "by oneself" or "on one's own."

Why is there no article before Mia nonna? Shouldn't it be La mia nonna?

In Italian, you drop the definite article (il or la) when talking about a singular family member with a possessive. So it is just mia nonna. If you were talking about multiple grandmothers, you would bring the article back: le mie nonne.

Could I say Mia nonna è vecchia instead of anziana?

Grammatically yes, but it sounds quite rude! Anziana is the polite and respectful word for "elderly." Calling a person vecchia can sound harsh, so it is usually saved for objects, like a vecchia macchina (old car).

If I am talking about my grandfather living alone, does da sola change?

Yes, the word sola agrees with the person you are talking about. For a grandfather, it becomes masculine: da solo. If your grandparents lived on their own together, it would be plural: da soli.