I'm interested in what linear/multilinear algebra does one need to study algebraic geometry(following EGA and Harthshorne). Texts I have in mind are like "Foundations of algebraic geometry" by Ravi Vakil.(If you know how much linear algebra this special book assumes, feel free to tell). I know you need abstract algebra to tackle algebraic geometry, specifically, ring theory, category theory and some homological algebra.
I'm not interested in classical algebraic geometry, that is, pre-scheme-theoretic algebraic geometry, thought I understand some think it is a natural way to start with this.
It depends on what you call linear algebra : usually this means working over a field.
To learn algebraic geometry, you will need a reasonably solid background in commutative algebra, meaning (commutative) rings and modules, including some basics of homological algebra.
You should at least know about modules and algebras, tensor products, quotients, localization. Being at ease with the language of categories is also important.