# Math 3 example statements

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 spaces: For a topological space \(X\), if \({F_1, F_2, F_3, \ldots}\) is a countable collection of open sets that are 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 \(f: \mathbb{R}^m \to \mathbb{R}^n\) 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 \(\displaystyle \frac{d(y_1, y_2, \ldots, y_n)}{d(x_1, x_2, \ldots, x_m)}\).

Parents:

- Math 3
Can you read the sort of things that professional mathematicians read, aka LaTeX formulas with a minimum of explanation?