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 \(\displaystyle \frac{d(y_1, y_2, \ldots, y_n)}{d(x_1, x_2, \ldots, x_m)}\).


  • Math 3

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