Hello mathchads, first-year physics scrub here. What are some time-saving computation tricks or beautiful little pieces of math that you wish you had known earlier?
I want to know how more mathematically mature people would go about computing/solving some specific problems that regularly come up.
It can be as trivial as Horner's method or even multiplying one side of an equation by a fraction equal to one to get rid of a variable.


My contributions to this thread:

Using Householder transformation to get the reflection matrix for any hyperplane intersecting the origin based on a vector orthogonal to it:
$H = I - \frac{2}{v^Tv}vv^T = I - 2vv^T\text{ for }||v||=1$
Here's simple derivation for a normalized vector: youtube.com/watch?v=6TIVIw4B5VA

Using Cramer's rule to solve linear systems of equations in two variables in your head, i.e. $x_1=\frac{a_{22}b_1-a_{12}b_2}{a_{11}a_{22}-a_{12}a_{21}}$