📐 Gram-Schmidt Procedure If $v_1, \\ldots, v_m$ is a linearly independent list in $V$, then there exists an orthonormal list $e_1, \\ldots, e_m$ such that $\\operatorname{span}(v_1, \\ldots, v_j) = \\operatorname{span}(e_1, \\ldots, e_j)$ for each $j$. From: linalg-axler Learn more:

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📖 Norm and Trace For a finite extension $K/F$ and $\\alpha \\in K$, the **norm** $N_{K/F}(\\alpha) = \\det(L_\\alpha)$ and the **trace** $T_{K/F}(\\alpha) = \\operatorname{tr}(L_\\alpha)$, where $L_\\alpha: K \\to K$ is left multiplication by $\\alpha$. From: gal-morandi Learn more:

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📐 Solution by Radicals (Full Version) Let $f(x) = 0$ be an equation with coefficients in a field $K$ (obtained from $\\mathbb{Q}$ by a finite number of adjunctions). A solution by radicals is a sequence of field extensions $K \\subset K_1 \\subset \\cdots \\subset K_\\mu$ where each $K_i$ is obtained by adjoining a $p_i$th root of an element of $K_{i-1}$ (with $p_i$th roots of unity present in $K_{i-1}$). Such a solution exists if ... From: gal-edwards Learn more:

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📖 Simple Group A group $G$ is called simple if it has no normal subgroups other than $\\{e\\}$ and $G$ itself. A simple group of order greater than 1 is an obstruction to solvability: if it appears as a quotient in any composition series, and it is not of prime order, then the group is not solvable. The alternating group $A_5$ (with 60 elements) is the smallest non-abelian simple group. From: gal-edwards Learn more:

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🎮 Interactive: Riemann Sum Visualizer Approximate area under a curve with rectangles. Watch how the limit of Riemann sums defines the integral. From: Calculus I Try it:

Calculus | Math Academy

An interactive course based on Tom M. Apostol

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🎮 Interactive: Eigenvalue Explorer Find vectors that only get scaled by a matrix. Eigenvalues reveal the fundamental structure of linear transformations. From: Linear Algebra Try it:

Introduction to Linear Algebra | Math Academy

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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: gal-howie Learn more:

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📐 Primitive Element Theorem If $E/F$ is a finite separable extension, then $E = F(\\alpha)$ for some $\\alpha \\in E$. Proof: If $F$ is infinite, let $E = F(\\alpha, \\beta)$. Choose $c \\in F$ so that $\\gamma = \\alpha + c\\beta$ is a primitive element. This is possible since only finitely many values of $c$ fail. The key step is showing $\\beta \\in F(\\gamma)$ by analyzing gcd arguments with minimal polynomials. From: gal-weintraub Learn more:

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🎮 Interactive: Non-Euclidean Lines Demo Explore straight lines in hyperbolic geometry. In the Poincare disk, geodesics appear as circular arcs perpendicular to the boundary. From: Four Pillars of Geometry Try it:

The Four Pillars of Geometry | Magic Internet Math

Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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🎮 Interactive: Matrix Transformation Demo Watch how matrices transform the plane. Rotations, reflections, shears, and stretches are all matrix multiplications! From: Linear Algebra Try it:

Introduction to Linear Algebra | Math Academy

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📖 Constructible Number A real number is constructible (by ruler and compass) if it can be obtained from the rational numbers by a finite sequence of operations involving addition, subtraction, multiplication, division, and the extraction of square roots. Equivalently, it belongs to a field that can be reached from $\\mathbb{Q}$ by a tower of quadratic extensions. From: gal-edwards Learn more:

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📐 Sufficient Condition for Diagonalizability If $T \\in \\mathcal{L}(V)$ has $\\dim V$ distinct eigenvalues, then $T$ is diagonalizable. From: linalg-axler Learn more:

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📐 Characterization of Cyclic Extensions Let $F$ contain a primitive $n$th root of unity. Then $K/F$ is cyclic of degree $n$ if and only if $K = F(\\alpha)$ where $\\alpha^n \\in F$. From: gal-morandi Learn more:

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📐 Galois Group of Cyclotomic Extensions $\\mathrm{Gal}(\\mathbb{Q}(\\zeta_n)/\\mathbb{Q}) \\cong (\\mathbb{Z}/n\\mathbb{Z})^\\times$, where the isomorphism sends $\\sigma_a$ to $a \\bmod n$, with $\\sigma_a(\\zeta_n) = \\zeta_n^a$. From: gal-jacobson Learn more:

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📖 Trivial and Non-trivial Solutions A solution $x_1, x_2, \\ldots, x_n$ of a homogeneous system is called non-trivial if not all of the elements are zero. Otherwise it is called trivial. From: gal-artin Learn more:

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📖 Right Vector Space An additive abelian group $V$ over a field $F$ with scalar multiplication written on the right, $Aa$, satisfying: (1) $(A+B)a = Aa + Ba$, (2) $A(a+b) = Aa + Ab$, (3) $(Ab)a = A(ba)$, (4) $A1 = A$. From: gal-artin Learn more:

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📐 Structure of Finite Fields For every prime $p$ and positive integer $n$, there exists a unique (up to isomorphism) field $\\mathbb{F}_{p^n}$ with $p^n$ elements. Its multiplicative group is cyclic of order $p^n - 1$. From: gal-weintraub Learn more:

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📐 Characteristic is Prime The characteristic of a field is either $0$ or a prime number $p$. Proof: If $\\operatorname{char}(F) = n = ab$ with $1 < a, b < n$, then $(a \\cdot 1)(b \\cdot 1) = n \\cdot 1 = 0$. Since $F$ is a field (hence an integral domain), either $a \\cdot 1 = 0$ or $b \\cdot 1 = 0$, contradicting the minimality of $n$. From: gal-weintraub Learn more:

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📖 Primitive Root of Unity An element $\\epsilon$ is a primitive $n$th root of unity if $\\epsilon^n = 1$ and $\\epsilon$ has order exactly $n$. The roots of $x^n - 1$ are $1, \\epsilon, \\epsilon^2, \\ldots, \\epsilon^{n-1}$. From: gal-artin Learn more:

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📐 Characterization of Normal Subgroups A subgroup $H$ of a group $G$ is normal if and only if for every $S \\in H$ and every $T \\in G$, the element $T^{-1}ST$ is in $H$. Proof: A subgroup $H$ has the property that every presentation differs from the first by a single substitution if and only if for every $T$ in $G$ and every $S$ in $H$, $T^{-1}ST$ is in $H$. In other words, the subgroup is invariant under conjugation by elements of $G$. From: gal-edwards Learn more:

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