I do not understand the full theory of tensors, but have heard several things about them:
Linear transformations are (1,1)-tensors.
Metrics are (0,2)-tensors.
Quadratic forms and metrics look very alike.
When we do basis transformations on linear transformations, we do similar matrices $P^{-1}AP$ which keeps the determinant, while on quadratic curves we do congruence matrices $P^{\mathrm{T}}AP$ which multiplies the determinant by $|P|^2$. This suggests me about covariation.
Someone has told me that we actually have cubic forms or even more of them not taught in linear algebra courses, because they require the language of tensors. Yet we only have two slots for duality, and (1,1,1,...,1)-tensor doesn't make any sense. only by making it (0,n) makes it compatible.
Do we have a full theory about all these vague ideas? Can the theory help make linear algebra clearer and more to its essence?