💡 Proposition V.17 (Separando)
If magnitudes be proportional componendo, they will also be proportional separando.
From: Euclid's Elements
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💡 The Liberal Achievement
The 19th century saw unprecedented progress in freedom, prosperity, and knowledge. This was not despite liberalism but because of it—yet we have forgotten what made it possible.
From: The Road to Serfdom
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📖 Definition V.3 (Ratio)
A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
From: Euclid's Elements
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📖 Infinite Series
The series $\\sum_{n=1}^\\infty a_n$ converges to $S$ if the partial sums $s_N = \\sum_{n=1}^N a_n$ converge to $S$.
From: Real Analysis
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📐 Criterion for Invertibility
$[a]_n$ has an inverse if and only if $\\gcd(a, n) = 1$.
From: intro-discrete
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📖 Deterministic Finite Automaton
A DFA is $(Q, \\Sigma, \\delta, q_0, F)$: states, alphabet, transition function, start state, accepting states.
From: intro-discrete
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💡 The Principle of Competition
Competition is the only method by which complex economic activities can be adjusted to changing circumstances without coercive or arbitrary intervention. It dispenses with the need for "conscious social control" and gives individuals a chance to decide whether their prospects are sufficient.
From: The Road to Serfdom
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📐 Euler
If $\\gcd(a, n) = 1$, then $a^{\\varphi(n)} \\equiv 1 \\pmod{n}$.
Proof: **Proof Sketch:**
1. Let $\\{r_1, r_2, \\ldots, r_{\\varphi(n)}\\}$ be the reduced residue system mod $n$ (all integers coprime to $n$ in $[1, n-1]$).
2. Since $\\gcd(a, n) = 1$, the set $\\{a \\cdot r_1, a \\cdot r_2, \\ldots, a \\cdot r_{\\varphi(n)}\\}$ is also a reduced residue system mod $...
From: Cryptography Math
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📐 Common Notion 4
Things which coincide with one another are equal to one another.
From: Euclid's Elements
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📖 Linear Discriminant Analysis (LDA)
LDA models the distribution of $X$ within each class as normal with a class-specific mean $\\mu_k$ and common variance $\\sigma^2$, then uses Bayes' theorem to classify observations.
From: Intro to Statistical Learning
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📐 Spectral Theorem (Real Symmetric Matrices)
Every real symmetric matrix is orthogonally diagonalizable: $A = QDQ^T$ where $Q$ is orthogonal and $D$ is diagonal.
Proof: **Step 1: Eigenvalues are real.** If $Av = \\lambda v$ with $v \\neq 0$:
$$\\bar{\\lambda} \\bar{v}^T v = \\bar{v}^T \\bar{\\lambda} v = \\bar{v}^T A v = (A^T \\bar{v})^T v = (A\\bar{v})^T v = \\lambda \\bar{v}^T v$$
Since $\\bar{v}^T v = \\|v\\|^2 > 0$, we have $\\bar{\\lambda} = \\lambda$.
**S...
From: Advanced Linear Algebra
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📐 Eigenvalues as Roots
$\\lambda$ is an eigenvalue of $A$ if and only if $\\lambda$ is a root of the characteristic polynomial.
Proof: $\\lambda$ is an eigenvalue $\\iff$ $(A - \\lambda I)v = 0$ has a nonzero solution
$\\iff$ $A - \\lambda I$ is not invertible
$\\iff$ $\\det(A - \\lambda I) = 0$
$\\iff$ $\\det(\\lambda I - A) = 0$
$\\iff$ $\\lambda$ is a root of the characteristic polynomial.
From: Advanced Linear Algebra
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💡 Proposition III.30
To bisect a given circumference.
From: Euclid's Elements
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💡 Proposition VI.32
If two triangles having two sides proportional to two sides be placed together at one angle so that their corresponding sides are also parallel, the remaining sides of the triangles will be in a straight line.
From: Euclid's Elements
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📐 Uniqueness of Identity and Inverses
The identity element is unique. Each element has a unique inverse.
From: intro-discrete
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📐 Homogeneous System Solutions
The solution set of a homogeneous system $Ax = \\mathbf{0}$ is a subspace of $F^n$, called the null space of $A$.
Proof: Let $N = \\{x \\in F^n : Ax = \\mathbf{0}\\}$.
- $A \\cdot \\mathbf{0} = \\mathbf{0}$, so $\\mathbf{0} \\in N$.
- If $x, y \\in N$, then $A(x + y) = Ax + Ay = \\mathbf{0} + \\mathbf{0} = \\mathbf{0}$.
- If $x \\in N$ and $c \\in F$, then $A(cx) = c(Ax) = c \\cdot \\mathbf{0} = \\mathbf{0}$.
From: Advanced Linear Algebra
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📖 Hidden Layer
Hidden layers are intermediate layers between input and output. Deep networks have multiple hidden layers, enabling hierarchical feature learning.
From: Intro to Statistical Learning
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📖 Binary Quadratic Form
A binary quadratic form is a homogeneous polynomial $f(x, y) = ax^2 + bxy + cy^2$ with integer coefficients. Gauss writes it as $(a, b, c)$.
From: Disquisitiones Arithmeticae
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📖 Perfect Graph
A graph $G$ is perfect if $\\chi(H) = \\omega(H)$ for every induced subgraph $H$ of $G$.
From: Introduction to Graph Theory
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💡 Proposition I.1
On a given finite straight line to construct an equilateral triangle.
From: Euclid's Elements
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📖 Kernel and Image
For a linear transformation $T: V \\to W$: the kernel (null space) is $\\ker(T) = \\{v \\in V : T(v) = \\mathbf{0}\\}$ and the image (range) is $\\text{im}(T) = \\{T(v) : v \\in V\\}$.
From: Advanced Linear Algebra
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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: Men of Mathematics
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