When performing some job, I may need to use certain tools: a scrub for cleaning the bathroom; a knife for cutting vegetables; a hand drill for securing a shelf; etc. Let's suppose the job concerns problemsolving. I may reach for a high level mental model appropriate for the problem. When we arrive at a specialty problem, like in a field of mathematics, the tools become more specialized too.
I'm currently participating in Machine Learning Tokyo's もくもく reading group, discussing the textbook Mathematics for Machine Learning; the mathematical field of interest is linear algebra. Here are the tools I see that one needs to learn in order to complete exercises at the end of each chapter of the book:

linear algebra:
 Abelian group
 congruence class
 Bezout theorem
 matrix multiplication
 homogenous / inhomogenous systems
 Gaussian elimination method
 matrix inversion
 linear subspaces
 linear independence
 linear combination
 linear basis
 linear mappings
 dimensionality
 rank
 image / kernel
 endomorphism

analytic geometry
 inner product
 distance function
 angle between vectors
 dot product
 subspace
 orthogonal projection
 cannonical basis
 endomorphism
 image / kernel
 GramSchmidt method
 CauchySchwartz inequality
 vector rotation

matrix decompositions
 Laplace expansion
 Sarrus rule
 determinant
 eigenspace
 invertible
 diagnolizable
 singular value decomposition
 rank approximation
Notice that subspaces, images, kernels, endomorphisms and rank come up multiple times. These may be tools that require mastery if you're going to be skilled at solving problems. Also notice when you come across a tool that you're unfamiliar with – learning it may increase the number of problems you're able to solve.
Becoming an expert does not just mean being able to use these tools to solve problems. To be an expert, you must also teach others how to use the tools effectively in their efforts to solve problems.