Formalism isn't really the problem though. The problem is that it needs to be accompanied with geometrical explanations and intuition (especially for linear algebra).

Formalism is here to help you put words on intuition. Intuition without formalism is as useless as formalism without intuition.

In linear algebra in particular, people who avoid formalism at all cost tend to focus on obscure calculation results on matrices instead on their more geometrical counterparts on linear maps