🔗 Steinitz Exchange Lemma
Let $\\{v_1, \\ldots, v_m\\}$ span $V$ and let $\\{w_1, \\ldots, w_n\\}$ be linearly independent in $V$. Then $n \\leq m$, and after reordering, $\\{w_1, \\ldots, w_n, v_{n+1}, \\ldots, v_m\\}$ spans $V$.
Proof: We proceed by induction on $n$.
**Base case ($n = 1$):** Since $w_1 \\neq \\mathbf{0}$ and the $v_i$ span $V$, we can write $w_1 = \\sum a_i v_i$ with some $a_j \\neq 0$. After reordering, say $a_1 \\neq 0$. Then $v_1 \\in \\text{span}(w_1, v_2, \\ldots, v_m)$, so $\\{w_1, v_2, \\ldots, v_m\\}$ ...
From: Advanced Linear Algebra
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📐 Invertibility and Determinant
A square matrix $A$ is invertible if and only if $\\det(A) \\neq 0$.
Proof: **(⇒)** If $A$ is invertible, then $AA^{-1} = I$. By the product rule:
$$\\det(A)\\det(A^{-1}) = \\det(I) = 1$$
So $\\det(A) \\neq 0$.
**(⇐)** If $\\det(A) \\neq 0$, then the columns of $A$ are linearly independent (if dependent, two columns would be proportional, giving det = 0). Thus $A$ has f...
From: Advanced Linear Algebra
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💡 Proposition VI.15
In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal.
From: Euclid's Elements
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📖 Quadratic Residue
An integer $a$ with $\\gcd(a, p) = 1$ is a quadratic residue (QR) modulo $p$ if $x^2 \\equiv a \\pmod{p}$ has a solution. Otherwise, $a$ is a quadratic non-residue (NR).
From: Disquisitiones Arithmeticae
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📖 Collectivism
Any system that organizes society to pursue collective goals chosen by central authority, deliberately subordinating individual purposes to the aims of the group. Socialism is merely one species of this broader genus.
From: The Road to Serfdom
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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: math_history
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💡 Proposition IV.9
About a given square to circumscribe a circle.
From: Euclid's Elements
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📖 Diophantine Equation
A Diophantine equation is a polynomial equation where we seek integer solutions.
From: Disquisitiones Arithmeticae
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💡 Proposition III.25
Given a segment of a circle, to describe the complete circle of which it is a segment.
From: Euclid's Elements
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💡 Proposition II.2
If a straight line be cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole.
From: Euclid's Elements
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🎮 Interactive: Determinant Visualizer
Understand the determinant as signed area/volume. See why det(AB) = det(A)det(B) and when matrices are invertible.
From: Linear Algebra
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📐 Integration of Uniform Limit
If $f_n \\to f$ uniformly on $[a,b]$, then $\\int_a^b f = \\lim \\int_a^b f_n$.
From: Real Analysis
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📐 Fermat
If $p$ is prime and $p \\nmid a$, then $a^{p-1} \\equiv 1 \\pmod{p}$.
Proof: Let $G_p = \\{[1], \\ldots, [p-1]\\}$. Multiplying all elements by $[a]$ permutes $G_p$. Product of $aG_p$ equals $[a]^{p-1}$ times product of $G_p$. Cancel to get $[a]^{p-1} = [1]$.
From: intro-discrete
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📖 Totalitarianism
A system where political power extends to the whole of life—where the state recognizes no autonomous spheres of individual action and all aspects of existence are subordinated to political ends.
From: The Road to Serfdom
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📖 Field
A field $(F, +, \\cdot)$ is a set $F$ with two binary operations satisfying: (1) $(F, +)$ is an abelian group with identity $0$, (2) $(F \\setminus \\{0\\}, \\cdot)$ is an abelian group with identity $1$, and (3) multiplication distributes over addition.
From: Advanced Linear Algebra
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📖 Index (Discrete Logarithm)
If $g$ is a primitive root modulo $p$ and $a \\equiv g^k \\pmod{p}$, then $k$ is called the index of $a$ to base $g$, written $\\text{ind}_g(a) = k$. This is the discrete logarithm.
From: Disquisitiones Arithmeticae
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📐 Cosets Partition the Group
The distinct left cosets of $H$ in $G$ partition $G$. All cosets have size $|H|$.
From: intro-discrete
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📖 Economic Planning
Central direction of all economic activity according to a single plan, laying down how society\
From: The Road to Serfdom
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📖 Coset
For $H \\leq G$ and $g \\in G$, the left coset is $gH = \\{gh : h \\in H\\}$.
From: intro-discrete
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📐 Cayley-Hamilton Theorem
Every square matrix satisfies its own characteristic polynomial: if $p(\\lambda) = \\det(\\lambda I - A)$, then $p(A) = 0$.
Proof: **Proof using Jordan form:** Over $\\mathbb{C}$, $A$ is similar to its Jordan form $J$. Since $p(A)$ and $p(J)$ are similar, it suffices to show $p(J) = 0$.
For a Jordan block $J_k(\\lambda)$, the characteristic polynomial is $(t - \\lambda)^k$.
We have $(J_k(\\lambda) - \\lambda I)^k = N^k$ wh...
From: Advanced Linear Algebra
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📐 Span is a Subspace
For any subset $S$ of a vector space $V$, $\\text{span}(S)$ is a subspace of $V$. Moreover, it is the smallest subspace containing $S$.
Proof: **Subspace:** Let $u = \\sum a_i v_i$ and $w = \\sum b_j u_j$ be in $\\text{span}(S)$.
- $u + w = \\sum a_i v_i + \\sum b_j u_j$ is a linear combination, so $u + w \\in \\text{span}(S)$.
- For $c \\in F$: $c \\cdot u = c \\sum a_i v_i = \\sum (ca_i) v_i \\in \\text{span}(S)$.
**Smallest:** Any s...
From: Advanced Linear Algebra
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📐 Unity of Economic and Political Freedom
Economic control is not control of a separate sector of life but control of the means for all our ends. Whoever has sole control of the means must also determine which ends are to be served.
Proof: Economic freedom means freedom to choose how we earn and spend our income. Without it, there is no freedom of the press (you need printing resources), no freedom of movement (you need transportation), no effective freedom at all. All freedoms require economic means for their exercise.
From: The Road to Serfdom
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