📖 Pointwise Convergence $(f_n)$ converges pointwise to $f$ if $\\forall x, \\forall \\varepsilon > 0, \\exists N: n > N \\Rightarrow |f_n(x) - f(x)| < \\varepsilon$. From: Real Analysis Learn more:

Real Analysis | Math Academy

A rigorous introduction to real analysis covering limits, continuity, differentiation, and integration through formal mathematical proofs.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Rank-Nullity Theorem (Dimension Theorem) For an $m \\times n$ matrix $A$ with rank $r$: $\\dim(C(A)) + \\dim(N(A)) = n$, or equivalently, $r + (n - r) = n$. Proof: The rank $r$ equals the number of pivot columns. The nullity (dimension of $N(A)$) equals the number of free variables, which is $n - r$. Thus $r + (n - r) = n$. From: Linear Algebra Learn more:

Introduction to Linear Algebra | Math Academy

Interactive course based on Gilbert Strang

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Null Space is a Subspace The null space of a linear transformation $T: V \\to W$ is a subspace of $V$. From: calc2 Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Euclid If $p = 2^n - 1$ is prime (Mersenne prime), then $2^{n-1} \\cdot p$ is perfect. Proof: Let $N = 2^{n-1}(2^n - 1)$ where $p = 2^n - 1$ is prime. Sum of divisors of $N$ is $(1 + 2 + \\cdots + 2^{n-1})(1 + p) = (2^n - 1)(2^n)$. $= 2^n(2^n - 1) = 2 \\cdot 2^{n-1}(2^n - 1) = 2N$. Sum of proper divisors = $2N - N = N$, so $N$ is perfect. From: numbers-geometry Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Integration of sin²(x) Using the identity $\\sin^2(x) = \\frac{1 - \\cos(2x)}{2}$, we have $\\int \\sin^2(x) \\, dx = \\frac{x}{2} - \\frac{\\sin(2x)}{4} + C$. From: Beginner Calculus Learn more:

Introductory Calculus | Math Academy

Calculus Made Easy - An interactive introduction to calculus

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

💡 Equilateral Triangle Construction Given segment $AB$, an equilateral triangle can be constructed with $AB$ as one side. Proof: Draw circle centered at $A$ with radius $AB$. Draw circle centered at $B$ with radius $BA$. Let $C$ be an intersection point of these circles. Then $AC = AB$ (radius of first circle) and $BC = BA$ (radius of second). Therefore $AB = BC = CA$, so triangle $ABC$ is equilateral. From: numbers-geometry Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Rank-Nullity Theorem For a linear transformation $T: V \\to W$ where $V$ is finite-dimensional: $\\dim(V) = \\text{rank}(T) + \\text{nullity}(T)$. Proof: Let $\\dim(V) = n$ and $\\dim(\\ker(T)) = k$. Let $\\{v_1, \\ldots, v_k\\}$ be a basis for $\\ker(T)$. Extend this to a basis $\\{v_1, \\ldots, v_k, u_1, \\ldots, u_{n-k}\\}$ of $V$. **Claim:** $\\{T(u_1), \\ldots, T(u_{n-k})\\}$ is a basis for $\\text{im}(T)$. **Spanning:** Any $w \\in \\text... From: Advanced Linear Algebra Learn more:

Advanced Linear Algebra | Magic Internet Math

A rigorous treatment of linear algebra based on Hoffman & Kunze, covering vector spaces, linear transformations, and canonical forms.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Theorem 11.9.1 (First Fundamental Theorem of Calculus) Let $f : [a, b] \\to \\mathbb{R}$ be Riemann integrable, and define $F(x) := \\int_a^x f$. Then $F$ is continuous on $[a, b]$. Moreover, if $f$ is continuous at $x_0 \\in (a, b)$, then $F$ is differentiable at $x_0$ with $F\ Proof: **Continuity:** For $x < y$, $|F(y) - F(x)| = |\\int_x^y f| \\leq M(y - x)$ where $M$ bounds $|f|$. As $y \\to x$, $F(y) \\to F(x)$. **Differentiability:** Fix $x_0$ where $f$ is continuous. For $h > 0$: $\\frac{F(x_0 + h) - F(x_0)}{h} = \\frac{1}{h} \\int_{x_0}^{x_0 + h} f$ Since $f$ is conti... From: tao-analysis-1 Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

🎮 Interactive: Polynomial Grapher Plot polynomials and find their roots. See how the degree determines the shape of the curve. From: Basic Algebra Try it:

Basic Algebra - Interactive Learning

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Projection Theorem If $S$ is a closed subspace of an inner product space $V$ and $v \\in V$, there exists a unique $s \\in S$ such that $\\|v - s\\| = \\inf\\{\\|v - w\\| : w \\in S\\}$. Moreover, $v - s \\perp S$. Proof: The infimum is achieved by completeness of $S$. Uniqueness follows from the parallelogram law. The orthogonality condition characterizes the projection. From: adv_linalg Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Cycle Decomposition Every permutation can be written as a product of disjoint cycles, unique up to order. From: aa-pinter Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Axiom 3.1 (Sets are Objects) If $A$ is a set, then $A$ is also an object. In particular, given two sets $A$ and $B$, it is meaningful to ask whether $A$ is also an element of $B$. From: tao-analysis-1 Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 First Isomorphism Theorem for Modules If $\\phi: M \\to N$ is an $R$-module homomorphism, then $M/\\ker(\\phi) \\cong \\text{Im}(\\phi)$. From: df-course Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Cyclic Shift A cyclic shift of $(c_0, c_1, \\ldots, c_{n-1})$ is $(c_{n-1}, c_0, \\ldots, c_{n-2})$. From: intro-discrete Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Order of an Element The \\textbf{order} of an element $a \\in G$, denoted $|a|$ or $\\text{ord}(a)$, is the smallest positive integer $n$ such that $a^n = e$. If no such $n$ exists, $a$ has \\textbf{infinite order}. From: df-course Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 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: tao2 Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Graph A graph $G$ is a triple $(V(G), E(G), \\psi_G)$ where $V(G)$ is a set of vertices, $E(G)$ is a set of edges, and $\\psi_G$ associates each edge with an unordered pair of vertices. A simple graph has no loops or multiple edges. From: Introduction to Graph Theory Learn more:

Introduction to Graph Theory | Math Academy

Interactive course based on Douglas B. West

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Rank-Nullity Theorem For a linear transformation $T: V \\to W$ with $V$ finite-dimensional, $\\dim(V) = \\dim(\\ker T) + \\dim(\\mathrm{im}\\, T)$. Proof: Let $\\{v_1, \\ldots, v_k\\}$ be a basis for $\\ker T$ and extend to a basis $\\{v_1, \\ldots, v_k, u_1, \\ldots, u_r\\}$ for $V$. Then $\\{Tu_1, \\ldots, Tu_r\\}$ is a basis for $\\mathrm{im}\\, T$, giving $\\dim V = k + r$. From: adv_linalg Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Finite-Dimensional One-to-One Characterization Let $T: V \\to W$ be linear with $\\dim V = n$. The following are equivalent: (a) $T$ is one-to-one. (b) If $e_1, \\ldots, e_p$ are independent in $V$, then $T(e_1), \\ldots, T(e_p)$ are independent. (c) $\\dim T(V) = n$. (d) If $\\{e_1, \\ldots, e_n\\}$ is a basis for $V$, then $\\{T(e_1), \\ldots, T(e_n)\\}$ is a basis for $T(V)$. From: calc2 Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Pascal Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel From: Men of Mathematics Learn more:

Men of Mathematics | Math Academy

An interactive journey through 2500 years of mathematical history with E.T. Bell

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.