Maybe I'm biased, but I really don't think so. Monads are quite a bit more abstract than the concepts in linear algebra. Linear algebra is both geometric and algorithmic and therefore very intuitive. Most of the difficulty people have learning linear algebra can be attributed to poor teaching methods.
That depends on the part of linear algebra. In an abstract function space when you start calculating dimensions of kernels and the like and get ready to make the jump to infinite dimensions, Banach spaces, and Hilbert spaces, it's about as abstract as monads.
Well, to be fair, functional analysis is not part of linear algebra proper. (If you want to get more abstract, you go to rings and modules and from there to category theory.)
Typically, linear algebra is understood to be the study of finite-dimensional vector spaces, so functional analysis is not necessarily part of it.
However, things like the vector space of polynomials of degree at most n, the vector space of all homomorphisms between two vector spaces, the dual space of a vector space, etc. are all concepts that belong to linear algebra proper yet are more "abstract" than just "computations with matrices".
