📐 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: Algebraic Number Theory Learn more:

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📐 Structure Theorem for Finitely Generated Abelian Groups Every finitely generated abelian group is isomorphic to $\\mathbb{Z}^r \\oplus \\mathbb{Z}/d_1\\mathbb{Z} \\oplus \\cdots \\oplus \\mathbb{Z}/d_k\\mathbb{Z}$ with $d_1 \\mid d_2 \\mid \\cdots \\mid d_k$. From: df-course Learn more:

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📖 Algebraic Extension An extension $K/F$ is \\textbf{algebraic} if every element of $K$ is algebraic over $F$. From: df-course Learn more:

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📖 Surjective (Onto) A function $f: A \\to B$ is surjective if for every $b \\in B$, there exists $a \\in A$ with $f(a) = b$. From: Real Analysis Learn more:

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📐 Euler Every rotation of $\\mathbb{R}^3$ about the origin is a rotation about some axis through the origin. From: Four Pillars of Geometry Learn more:

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💡 Proposition VII.1 Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until a unit is left, the original numbers will be prime to one another. From: Euclid's Elements Learn more:

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📖 Exact Sequence A sequence $\\cdots \\to M_{i-1} \\xrightarrow{\\phi_{i-1}} M_i \\xrightarrow{\\phi_i} M_{i+1} \\to \\cdots$ is \\textbf{exact} at $M_i$ if $\\text{Im}(\\phi_{i-1}) = \\ker(\\phi_i)$. From: df-course Learn more:

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💡 Proposition I.11 To draw a straight line at right angles to a given straight line from a given point on it. From: Euclid's Elements Learn more:

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📖 Support Vector Machine SVM extends the support vector classifier using kernels to handle non-linear boundaries: $f(x) = \\beta_0 + \\sum_{i \\in S}\\alpha_i K(x, x_i)$ where $K$ is a kernel function. From: Intro to Statistical Learning Learn more:

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📐 Row Equivalence Theorem Row equivalent matrices have the same row space and the same null space. From: Advanced Linear Algebra Learn more:

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📐 Matrix Multiplication as Composition For linear transformations $T: V \\to W$ and $S: W \\to X$ with appropriate bases: $[S \\circ T]_\\beta^\\delta = [S]_\\gamma^\\delta [T]_\\beta^\\gamma$. Proof: Let $\\{v_1, \\ldots, v_n\\}$ be the basis $\\beta$ of $V$. For each $j$: $$[(S \\circ T)(v_j)]_\\delta = [S(T(v_j))]_\\delta = [S]_\\gamma^\\delta [T(v_j)]_\\gamma = [S]_\\gamma^\\delta ([T]_\\beta^\\gamma)_j$$ where $([T]_\\beta^\\gamma)_j$ denotes column $j$ of $[T]_\\beta^\\gamma$. This equ... From: Advanced Linear Algebra Learn more:

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📖 Residual Standard Error (RSE) $\\text{RSE} = \\sqrt{\\frac{1}{n-2}\\text{RSS}} = \\sqrt{\\frac{1}{n-2}\\sum_{i=1}^{n}(y_i - \\hat{y}_i)^2}$ From: Intro to Statistical Learning Learn more:

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📖 ROC Curve and AUC The ROC curve plots sensitivity (true positive rate) vs. 1 - specificity (false positive rate) for all classification thresholds. AUC (area under the curve) summarizes overall classifier performance. From: Intro to Statistical Learning Learn more:

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📖 Analytic Function A function is analytic at $a$ if it equals its Taylor series in some neighborhood of $a$. From: Real Analysis Learn more:

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📐 Altitude to Hypotenuse In a right triangle, the altitude to the hypotenuse creates two triangles similar to the original and to each other. The altitude length is the geometric mean of the two segments it creates on the hypotenuse. Proof: Let $\\triangle ABC$ have right angle at $C$, with altitude $CD$ to hypotenuse $AB$. $\\triangle ACD \\sim \\triangle ABC$ (AA similarity: shared angle at $A$, right angles). $\\triangle BCD \\sim \\triangle ABC$ (AA similarity: shared angle at $B$, right angles). Therefore $\\triangle ACD \\sim ... From: Four Pillars of Geometry Learn more:

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📐 Bolzano-Weierstrass Theorem Every bounded sequence has a convergent subsequence. From: Real Analysis Learn more:

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📖 Hazard Function $h(t) = \\lim_{\\Delta t \\to 0}\\frac{\\Pr(t < T \\leq t + \\Delta t | T > t)}{\\Delta t}$ is the instantaneous rate of event occurrence given survival to time $t$. From: Intro to Statistical Learning Learn more:

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📐 Uniqueness of Identities and Inverses The additive identity 0 and multiplicative identity 1 are unique. Additive and multiplicative inverses are unique. From: Real Analysis Learn more:

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📖 Completeness Axiom Every non-empty subset of $\\mathbb{R}$ that is bounded above has a least upper bound (supremum) in $\\mathbb{R}$. From: Real Analysis Learn more:

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💡 Proposition I.9 To bisect a given rectilineal angle. From: Euclid's Elements Learn more:

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