📐 Basis Extension Theorem
Every linearly independent set in a finite-dimensional vector space can be extended to a basis.
Proof: Let $S = \\{v_1, \\ldots, v_k\\}$ be linearly independent in $V$ with $\\dim(V) = n$.
If $\\text{span}(S) = V$, then $S$ is already a basis.
Otherwise, there exists $v_{k+1} \\in V \\setminus \\text{span}(S)$. Then $\\{v_1, \\ldots, v_k, v_{k+1}\\}$ is linearly independent.
Continue this proce...
From: Advanced Linear Algebra
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💡 Proposition IV.2
In a given circle to inscribe a triangle equiangular with a given triangle.
From: Euclid's Elements
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💡 Proposition II.14 (Quadrature)
To construct a square equal to a given rectilinear figure.
From: Euclid's Elements
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🎮 Interactive: Group Cayley Table Builder
Build multiplication tables for finite groups. Discover patterns in Z_n, symmetric groups, and dihedral groups.
From: Abstract Algebra
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🎮 Interactive: Binary Quadratic Form Explorer
Investigate which integers can be represented as ax^2 + bxy + cy^2. A key tool Gauss used to study primes of the form x^2 + ny^2.
From: Disquisitiones Arithmeticae
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📐 Differentiation of Limit
If $f_n \\to f$ pointwise, $f_n\
From: Real Analysis
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📐 Sum of First n Positive Integers
$1 + 2 + \\cdots + n = \\frac{n(n+1)}{2}$
Proof: Base case: $n=1$ gives $1 = 1(2)/2$. Induction step: Assume true for $k$. Then $1 + \\cdots + k + (k+1) = \\frac{k(k+1)}{2} + (k+1) = \\frac{(k+1)(k+2)}{2}$.
From: intro-discrete
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📖 Definition VII.4 (Parts)
But parts when it does not measure it.
From: Euclid's Elements
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💡 Proposition III.11
If two circles touch one another internally, and their centres be taken, the straight line joining their centres, if it be produced, will fall on the point of contact of the circles.
From: Euclid's Elements
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📖 Definition VII.19 (Cube Number)
A cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.
From: Euclid's Elements
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📐 Count of Quadratic Residues
For an odd prime $p$, exactly $(p-1)/2$ of the integers $1, 2, \\ldots, p-1$ are quadratic residues, and $(p-1)/2$ are non-residues.
From: Disquisitiones Arithmeticae
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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: probability
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📐 Five Color Theorem
Every planar graph is 5-colorable.
Proof: Induction on $n$. By the edge bound, some vertex $v$ has degree $\\leq 5$. Remove $v$, 5-color the rest by induction. If $d(v) < 5$ or two neighbors share a color, color $v$ with an available color. Otherwise, $v$ has 5 differently-colored neighbors. Consider a Kempe chain argument: swap colors a...
From: Introduction to Graph Theory
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📐 Change of Basis Formula
If $[T]_\\beta$ and $[T]_{\\beta\
Proof: Let $I: V \\to V$ be the identity map. Then $[I]_{\\beta'}^\\beta = P$ is the change of basis matrix.
We have $T = I \\circ T \\circ I$, so:
$$[T]_{\\beta'} = [I]_\\beta^{\\beta'} [T]_\\beta [I]_{\\beta'}^\\beta = P^{-1} [T]_\\beta P$$
From: Advanced Linear Algebra
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📏 Postulate 2
To produce a finite straight line continuously in a straight line.
From: Euclid's Elements
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📐 Chain Rule
If $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $(f \\circ g)\
From: Real Analysis
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📖 Relatively Prime
Two positive integers $a$ and $b$ are relatively prime (coprime) if $\\gcd(a, b) = 1$.
From: intro-discrete
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📐 Arzelà-Ascoli Theorem
A subset of $C([a,b])$ is precompact iff it is bounded and equicontinuous.
From: Real Analysis
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💡 The Security Trap
The more we seek absolute security through state provision, the more we sacrifice liberty. Ultimate security means complete dependence—and the "security" of the slave who is fed but not free.
From: The Road to Serfdom
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📐 Comparison Test
If $0 \\leq a_n \\leq b_n$ and $\\sum b_n$ converges, then $\\sum a_n$ converges. If $\\sum a_n$ diverges, so does $\\sum b_n$.
From: Real Analysis
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