Math 3 example statements

by Joe Zeng Jul 7 2016 updated Jul 10 2016

If you can read these formulas, you're in Math 3!

If you're at a Math 3 level, you'll probably be familiar with at least some of these sentences and formulas, or you would be able to understand what they meant on a surface level if you were to look them up. Note that you don't necessarily have to understand the proofs of these statements (that's what we're here for, to teach you what they mean), but your eyes shouldn't gloss over them either.

In a group $~$G$~$, the Conjugacy class of an element $~$g$~$ is the set of elements that can be written as $~$hgh^{-1}$~$ for all $~$h \in G$~$.

The [ rank-nullity theorem] states that for any [-linear_mapping] $~$f: V \to W$~$, the [-dimension] of the [-image] of $~$f$~$ plus the dimension of the [-kernel] of $~$f$~$ is equal to the dimension of $~$V$~$.

A [ Baire space] is a space that satisfies [ Baire's Theorem] on [complete_metric_space complete metric spaces]: For a [-topological_space] $~$X$~$, if $~${F_1, F_2, F_3, \ldots}$~$ is a [countable_set countable] collection of open sets that are [dense_set dense] in $~$X$~$, then $~$\bigcap_{n=1}^\infty F_n$~$ is also dense in $~$X$~$.

The [riemann_hypothesis] asserts that every non-trivial zero of the [riemann_zeta_function] $~$\zeta(s) = \sum_{n=1}^\infty \frac{1}{s^n}$~$ when $~$s$~$ is a complex number has a real part equal to $~$\frac12$~$.

$~$\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}$~$ The [jacobian_matrix] of a [vector_valued_function vector-valued function] $~$f: \mathbb{R}^m \to \mathbb{R}^n$~$ is the matrix of [-partial_derivatives] $~$\left[ \begin{matrix} \pd{y_1}{x_1} & \pd{y_1}{x_2} & \cdots & \pd{y_1}{x_m} \\ \pd{y_2}{x_1} & \pd{y_2}{x_2} & \cdots & \pd{y_2}{x_m} \\ \vdots & \vdots & \ddots & \vdots \\ \pd{y_n}{x_1} & \pd{y_n}{x_2} & \cdots & \pd{y_n}{x_m} \end{matrix} \right]$~$ between each component of the argument vector $~$x = (x_1, x_2, \ldots, x_m)$~$ and each component of the result vector $~$y = f(x) = (y_1, y_2, \ldots, y_n)$~$. It is notated as $~$\displaystyle \frac{d\mathbf{y}}{d\mathbf{x}}$~$ or $~$\displaystyle \frac{d(y_1, y_2, \ldots, y_n)}{d(x_1, x_2, \ldots, x_m)}$~$.