December 25, 2019

mathematical tools

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 problem-solving. 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
    • Gram-Schmidt method
    • Cauchy-Schwartz 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.

Content by © Jared Davis 2019
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