Linear Algebra
Problems using linear algebra (rather than problems about linear algebra). Most of the problems here are combinatorial in nature as a result, and there is a mix of linear algebra over $\mathbb R$ and linear algebra over $\mathbb{F}_2$.
Content and Brief overview
This is one of the most technical units at OTIS and is a part of the "technical" class of the units at OTIS. The unit starts off with a suggestion to not treat Linear Algebra as a study of matrices (Small tip: Don't ignore the suggestion :p).
The unit starts off by discussing the core of Linear Algebra concerning vector spaces and Linear Maps. This leads to the first and a pretty important walkthrough that introduces the readers to Lagrange Interpolation. Following which is a discussion about Dot product and linearly independent vectors. A rather interesting Grobber's lemma is introduced shortly. Just for fun, The kirchoff's Matrix Theorem is stated with its wikipedia link attached to satisfy the inquisitive reader.
Pre-requisites
As said earlier, this is a pretty technical unit. Hence, it is assumed that the reader is fluent with basic combinatorial problem solving before getting started with this unit. It is recommended that one is familiar with Analyzing equality cases and is somewhat fluent in working with elementary Local and Global ideas.
Prominent Problems
- USA TST 2016 #5 (Features as a walkthrough)
- USAMO 2021 #3 (Again, features as a walkthrough)
- TSTST 2018 #2
- Brazil Revenge 2016 #5 (A 9 pointer. One of the hardest, if not the hardest problem, in the unit.)
- China 2019 #5