📐 Properties of Characteristic Polynomial If $A$ is an $n \\times n$ matrix, then $f(\\lambda) = \\det(\\lambda I - A)$ is a polynomial of degree $n$. The leading term is $\\lambda^n$ and the constant term is $f(0) = (-1)^n \\det A$. From: calc2-course Learn more:

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📖 The Number Plane $\\mathbb{R}^2$ The number plane $\\mathbb{R}^2$ is the set of all ordered pairs $(x, y)$ of real numbers, equipped with the Cartesian coordinate system. From: Four Pillars of Geometry Learn more:

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📖 BIP-44 Derivation Path Standard path: $m / 44\ From: bips Learn more:

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📖 Relation A relation on set $A$ is a subset $R \\subseteq A \\times A$. We write $a \\sim b$ if $(a, b) \\in R$. From: intro-discrete Learn more:

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📖 Homogeneous Coordinates A point in $\\mathbb{RP}^2$ is represented by homogeneous coordinates $[x : y : z]$, where $(x, y, z) \\neq (0, 0, 0)$ and $[x : y : z] = [\\lambda x : \\lambda y : \\lambda z]$ for any $\\lambda \\neq 0$. From: Four Pillars of Geometry Learn more:

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📐 Structure of Simple Algebraic Extensions If $\\alpha$ is algebraic over $F$ with minimal polynomial $m$ of degree $n$, then $F(\\alpha) \\cong F[x]/(m(x))$ and $[F(\\alpha):F] = n$. From: df-course Learn more:

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📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$ Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset. Therefore, every element of $A$ is in $B$. Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition. Therefore, every element of $B$ is in $A$. Since $A \\subseteq B$ and $B \\subseteq A... From: Thales to Euclid Learn more:

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Journey through 2,500 years of mathematical history, from ancient Egypt through modern foundations

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💡 Proposition I.17 In any triangle two angles taken together in any manner are less than two right angles. From: Euclid's Elements Learn more:

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📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$ Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset. Therefore, every element of $A$ is in $B$. Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition. Therefore, every element of $B$ is in $A$. Since $A \\subseteq B$ and $B \\subseteq A... From: zeus Learn more:

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💡 Proposition VI.10 To cut a given uncut straight line similarly to a given cut straight line. From: Euclid's Elements Learn more:

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💡 Proposition VI.16 If four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and, if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines will be proportional. From: Euclid's Elements Learn more:

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📐 Construction of Finite Fields $\\mathbb{F}_{p^n}$ is the splitting field of $x^{p^n} - x$ over $\\mathbb{F}_p$. From: df-course Learn more:

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📐 When is -1 a Quadratic Residue? $\\left(\\frac{-1}{p}\\right) = (-1)^{(p-1)/2} = \\begin{cases} 1 & \\text{if } p \\equiv 1 \\pmod{4} \\\\ -1 & \\text{if } p \\equiv 3 \\pmod{4} \\end{cases}$ From: Disquisitiones Arithmeticae Learn more:

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📐 Orthogonal Transformations A linear transformation $T: \\mathbb{R}^n \\to \\mathbb{R}^n$ preserves lengths and angles iff its matrix $A$ satisfies $A^TA = I$ (orthogonal matrix). Such transformations preserve the inner product. From: Four Pillars of Geometry Learn more:

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📖 Producers and Looters The distinction between those who create value (producers) and those who survive by seizing values from others through political manipulation or force (looters). From: Atlas Shrugged Learn more:

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📖 Logical Connectives Negation ($\\neg$), conjunction ($\\land$), disjunction ($\\lor$), implication ($\\Rightarrow$), biconditional ($\\Leftrightarrow$). From: intro-discrete Learn more:

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📖 Definition: Transpose The transpose of $A$, denoted $A^T$, is the matrix obtained by exchanging rows and columns: $(A^T)_{ij} = A_{ji}$. From: Linear Algebra Learn more:

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📐 Fundamental Theorem of Finite Abelian Groups Every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order: $G \\cong \\mathbb{Z}/p_1^{a_1}\\mathbb{Z} \\times \\cdots \\times \\mathbb{Z}/p_k^{a_k}\\mathbb{Z}$. This decomposition is unique up to order. From: df-course Learn more:

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💡 Proposition I.47 (Pythagorean Theorem) In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. From: Euclid's Elements Learn more:

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📖 Pythagorean Triples A Pythagorean triple $(a, b, c)$ is a set of positive integers satisfying $a^2 + b^2 = c^2$. Examples: $(3, 4, 5)$, $(5, 12, 13)$, $(8, 15, 17)$. From: Four Pillars of Geometry Learn more:

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