Exercises for the mathematics tutorial

This chapter contains exercises for the mathematics tutorial on the first day of the school. The focus is on linear algebra, with some set theory exercises as well, and then some exercises on continuity and differentiation. Start with linear algebra — these are the most important exercises. The book contains many more exercises as well, if you find time and inspiration!

A selection in PDF format can be found hare: PDF file link

The beginner list contains exercises that we consider to be especially useful for the students who feel that the need some refreshing to do before embarking on the remaining exercises.

The intermediate list contains more advanced exercises, accessible for the majority of the students at the school.

The for the curious list contains exercises that go deeper into either mathematics or applications, and can be useful for students that already master the mathematics material at an advanced level. That being said, they should be interesting for most students.

Beginner

• Exercise 3.1 — Very basic sums
• Exercise 3.2 — Simplifying as a sum
• Exercise 6.1 — Basic vector operations
• Exercise 6.2 — Basic matrix-vector products
• Exercise 6.3 — Matrix-matrix products
• Exercise 9.1 — Checking candidate eigenvectors
• Exercise 9.2 — Eigenvalues in chemical kinetics
• Exercise 10.1 — Cooking with vectors 1
• Exercise 1.1 — Days of the week as a set
• Exercise 1.9 — Classically allowed region for Morse potential
• Exercise 6.4 — Linear independence and basis
• Exercise 21.1 — A first discontinuity
• Exercise 21.3 — Continuity by direct substitution
• Exercise 21.11 — Continuity of a model potential energy
• Exercise 21.12 — First partial derivatives
• Exercise 21.13 — Partial derivatives as one-dimensional derivatives
• Exercise 21.15 — A first Jacobian

Intermediate

• Exercise 6.6 — Office coordinates
• Exercise 6.7 — Boat sailing on the ocean
• Exercise 6.8 — An exercise by Robert Beezer
• Exercise 6.9 — Compute the action of a matrix on vectors
• Exercise 6.10 — Matrix-matrix products, rows and columns
• Exercise 9.3 — Eigenvalues and diagonalization of a Hermitian matrix
• Exercise 9.4 — A matrix that cannot be diagonalized
• Exercise 10.2 — Cooking with vectors 2
• Exercise 1.7 — De Morgan’s laws
• Exercise 1.10 — Molecules as sets
• Exercise 27.3 — The exponential, sine, and cosine series

For the curious

• Exercise 8.1 — Hybrid orbitals from vectors and matrices
• Exercise 9.6 — Eigenvalues of a reaction network
• Exercise 4.4 — Stirling’s approximation
• Exercise 5.12 — A finite-dimensional quantum-mechanical impossibility
• Exercise 11.1 — Complex numbers as a real vector space
• Exercise 21.21 — Checking multivariable limits along different paths
• Exercise 21.23 — Explore a two-dimensional energy landscape