How deep should ones linear algebra knowledge be before starting data science?
I think it truly depends on what you decide to specialize in. Data science is a very broad field, and you can actually work with data without knowing what eigenvalues and eigenvectors are. However, if you want to acquire a intermediate/advanced understanding of statistics or machine learning, you need at least an intermediate/advanced knowledge of linear algebra.

I suggest to take an introductory class on linear algebra on MOOC - just to have a more precise idea of what linear algebra is - and then study some other topics that you are interested in. Linear algebra is a useful tool, but it can be very boring, especially if you are an "applied" kind of guy. Moreover, I think that some concepts like eigenvalues or eigenvectors are easier to understand when seen in an applied context, e.g. principal component analysis.