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?

    • Mathematics

      Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.