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text: '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.\n\n> In a [3gd group] $G$, the [-4bj] of an element $g$ is the set of elements that can be written as $hgh^{-1}$ for all $h \\in G$.\n\n> 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 $V$.\n\n> 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 $X$.\n\n> 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 $\\frac12$.\n\n> $\\newcommand{\\pd}[2]{\\frac{\\partial #1}{\\partial #2}}$ 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 $\\displaystyle \\frac{d(y_1, y_2, \\ldots, y_n)}{d(x_1, x_2, \\ldots, x_m)}$.',
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