A
- abelian (1)
- abelian group (49)
- action (1)
- action by conjugation (1)
- addition theorem of trigonometric functions (1)
- additive group (7)
- adjoint matrix (2)
- algebraically closed field (1)
- algebraic element (1)
- algebraic extension (1)
- algebraic integer (1)
- algebraic multiplicity (22)
- algebraic number (1)
- alternating group (1)
- annihilate (1)
- annihilator (2)
- annihilator ideal (2)
- Artinian (1)
- Artinian ring (1)
- Artin–Schreier theory (1)
- ascending chain condition (1)
- ascending chain of ideals (1)
- ascending chain of submodules (2)
- augmentation ideal (1)
- augmentation map (1)
- augmented matrix (30)
- automorphism (9)
- automorphism group (2)
- axiom (1)
- axiom of group (1)
- axiom of probability (1)
B
- back-substitution (1)
- ball (2)
- basis (76)
- basis for a vector space (31)
- basis of a vector space (23)
- basis vector (1)
- Bayes’ rule (2)
- Bayes’rule Bayes’ theorem (1)
- Bayes’s rule (1)
- Bayes’s theorem (1)
- Bayes’ theorem (2)
- Bays’ rule (1)
- Bays’s theorem (1)
- Berkeley (9)
- Berkeley.LA (9)
- Bernoulli distribution (1)
- Bernoulli random variable (2)
- bijective (3)
- bijective homomorphism (1)
- binomial coefficient (1)
- binomial expansion (2)
- binomial random variable (1)
- binomial theorem (2)
- block matrix (1)
- Boolean ring (2)
- Boston College (1)
- Boston College.LA (1)
- box (1)
C
- calculus (1)
- cancellation law (1)
- cards (1)
- category of abelian groups (1)
- category of rings (1)
- category theory (4)
- Cauchy-Schwarz inequality (1)
- Cayley-Hamilton theorem (14)
- center (5)
- center of a group (10)
- center of a ring (1)
- centralizer (6)
- change of basis (2)
- change of coordinates (1)
- characteristic (1)
- characteristic equation (2)
- characteristic group (2)
- characteristic polynomi (1)
- characteristic polynomial (61)
- characteristic polynomials (1)
- characteristic subgroup (2)
- Chebyshev’s inequality (1)
- circle (1)
- circulant matrix (1)
- circumference (1)
- class equation (1)
- coefficient matrix (2)
- cofactor (1)
- cofactor expansion (19)
- coin (4)
- coin toss (3)
- colon ideal (1)
- column (1)
- column rank (1)
- column space (5)
- column vector (6)
- comaximal (1)
- comaximal ideal (1)
- common eigenvalues (1)
- common eigenvector (4)
- communication network (1)
- commutative ring (36)
- commutator (9)
- commutator subgroup (6)
- companion matrix (3)
- complement (3)
- complex conjugate (6)
- complex eigenvalue (6)
- complex matrix (2)
- complex numbers (2)
- complex vector (2)
- composite (1)
- composite of functions (1)
- composition series (2)
- conditional probability (13)
- congruence (2)
- conjugacy class (2)
- conjugate (10)
- conjugate transpose (1)
- conjugation (2)
- consistent system (7)
- constant vector (1)
- continued fraction (1)
- continuous function (6)
- continuous random variable (1)
- coordinate vector (16)
- coordinate vectors (2)
- cos (2)
- coset (2)
- cosine (2)
- cosine function (5)
- counterexample (10)
- coupon collecting problem (1)
- Cramer’s Rule (1)
- cross product (3)
- cubic polynomial (2)
- cyclic Galois group (1)
- cyclic group (24)
- cyclic module (2)
- cyclotomic extension (2)
- cyclotomic field (4)
D
- decreasing sequence of events (1)
- defective matrix (3)
- defective rate (2)
- defending chain condition on ideals (1)
- degree (1)
- degree of field extension (3)
- derivative (12)
- derivative linear transformation (1)
- descending chain of ideals (1)
- determinant (48)
- determinant of a matrix (34)
- diagonal (1)
- diagonal entry (1)
- diagonalizable (18)
- diagonalizable matrix (18)
- diagonalization (27)
- diagonalization of a linear transformation (1)
- diagonalization of a matrix (11)
- diagonalize a matrix (2)
- diagonal matrix (33)
- diagonal subgroup (1)
- diagram (1)
- diametor (1)
- dice (3)
- die (3)
- differential equation (2)
- differentiation (5)
- dihedral group (3)
- dimension (23)
- dimension of a vector space (16)
- direct product (7)
- direct product of groups (4)
- direct product of ideals (1)
- direct product of rings (2)
- direct sum (2)
- discrete random variable (1)
- discriminant (1)
- distance (3)
- distribution law (1)
- divisibility (1)
- divisible (1)
- divisible group (1)
- Division Algorithm (2)
- domain (6)
- dominant eigenvalue (1)
- dot product (22)
- dual basis (1)
- dual vector space (1)
- Dummit & Foote (1)
E
- echelon form (6)
- echelon from (1)
- Eckmann–Hilton argument (1)
- eigenbasis (3)
- eigenspace (25)
- eigenvalue (126)
- eigenvalues (5)
- eigenvecgtor (1)
- eigenvector (81)
- Eisenstein (1)
- Eisenstein’s criterion (13)
- Eisenstein integer (1)
- Eisenstein polynomial (8)
- Eisenstein prime (1)
- elementary number theory (1)
- elementary row operation (5)
- elementary row operations (35)
- entrance exam (1)
- epi (1)
- equivalence relation (1)
- Euclid (1)
- Euclidean Domain (2)
- Euclidean space (1)
- Euler (2)
- Euler’s formula (1)
- Euler totient function (1)
- even perfect number (1)
- event (4)
- exam (104)
- example (4)
- expectation (7)
- expected value (8)
- exponential distribution (1)
- exponential function (5)
- exponential random variable (1)
- extension degree (2)
- external semidirect product (1)
F
- factor group (2)
- Fermat’s Little Theorem (2)
- Fermat number (1)
- Fermat prime (1)
- field (25)
- field automorphism (1)
- field extension (17)
- field homomorphism (2)
- field norm (1)
- field of characteristic 0 (1)
- field of characteristic p (3)
- field of rational numbers (1)
- field theory (27)
- figure (2)
- filtration (1)
- final exam (1)
- finite abelian group (1)
- finite dimensional vector space (2)
- finite field (7)
- finite group (33)
- finitely generated (1)
- finitely generated abelian group (3)
- finitely generated module (1)
- finite order (1)
- finite ring (1)
- finite set (2)
- first isomorphism theorem (11)
- form a triangle (1)
- formula for pi (1)
- fourth isomorphism theorem (3)
- free action (1)
- free rank (1)
- free variable (2)
- Frobenius-Perron eigenvalue (1)
- Frobenius endomorphism (1)
- full rank (1)
- fundamental theorem of abelian groups (1)
- fundamental theorem of algebra (1)
- fundamental theorem of finitely generated abelian group (1)
G
- Galois extension (1)
- Galois group (3)
- Galois theory (3)
- game (1)
- Gauss (1)
- Gauss-Jorda (1)
- Gauss-Jordan elimination (20)
- gaussian distribution (1)
- Gaussian elimination (3)
- Gaussian integer (1)
- Gaussian integers (2)
- Gaussian integral (1)
- Gaussian prime (1)
- general formula (1)
- general linear group (1)
- general vec (1)
- general vector space (7)
- generator (14)
- generator of an ideal (1)
- generators of a group (1)
- geometric distribution (3)
- geometric multiplicity (17)
- geometric progression (1)
- geometric random variable (3)
- Gram-Schmidt orthogonalization process (4)
- Gram-Schmidt process (4)
- graph (1)
- greatest common divisor (1)
- group (82)
- group action (9)
- group automorphism (3)
- group homomorphism (35)
- group isomorphism (5)
- group of integers (2)
- group of order 12 (1)
- group of order 24 (1)
- group of rational numbers (4)
- group operation (8)
- group ring (1)
- group theory (126)
H
- harmonic sum (1)
- Harvard (5)
- Harvard.LA (5)
- Heisenberg group (1)
- hermitian matrix (8)
- high school (1)
- homogeneous equation (1)
- homogeneous system (15)
- homomorphism (48)
- how to compute determinant (1)
- how to find inverse matrix (2)
- hyperplane (2)
I
- ideal (43)
- ideal quotient (1)
- idempotent (13)
- idempotent element (1)
- idempotent matrix (10)
- identity element (3)
- identity matrix (6)
- image (6)
- image of a group homomorphism (3)
- image of a linear transformation (1)
- image of a ring homomorphism (1)
- imaginary number (2)
- inconsistent system (7)
- increasing sequence of events (1)
- independent (3)
- independent events (5)
- independent random variable (1)
- index (6)
- index 2 (2)
- indicator variable (1)
- induced homomorphism (1)
- induction (5)
- inequality (1)
- infinite extension (1)
- infinite field (1)
- infinite group (1)
- infinitely many prime numbers (1)
- infinitely many solutions (2)
- infinite sequence of events (1)
- infinite series (1)
- infinitude of primes (1)
- initial object (1)
- initial value (1)
- injective (5)
- injective homomorphism (7)
- injective linear transformation (3)
- inner product (23)
- inner product space (3)
- integer matrix (2)
- integral (2)
- integral by parts (1)
- integral domain (16)
- integration (1)
- interchangeable (1)
- internal semidirect product (1)
- intersection (1)
- intersection of groups (1)
- intersection of subspaces (1)
- intuitive (1)
- invariant factor (1)
- invariant factor decomposition (1)
- inverse (1)
- inverse element (6)
- inverse element in a group (1)
- inverse linear transformation (1)
- inverse map (1)
- inverse matrix (51)
- inverse matrix of a 2 by 2 matrix (1)
- invertible (4)
- invertible element (1)
- invertible matrix (57)
- irrational number (1)
- irreducible (2)
- irreducible element (4)
- irreducible ideal (1)
- irreducible module (2)
- irreducible polynomial (18)
- isomorphic (2)
- isomorphism (21)
- isomorphism of groups (2)
- isomorphism of vector spaces (6)
- isomorphism theorem (20)
J
- jewelry (1)
- Johns (1)
- Johns Hopkins (3)
- Johns Hopkins.LA (3)
- joint density function (1)
- joint probability mass function (1)
- Jordan canonical form (7)
K
- kernel (41)
- kernel of (1)
- kernel of a group homomorphism (11)
- kernel of a linear transformation (4)
- kernel of a matrix (11)
- kernel of a ring homomorphism (4)
- Kyoto (3)
- Kyoto.LA (3)
- Kyushu (1)
- Kyushu.LA (1)
L
- Lagrange’s theorem (8)
- largest eigenvalue (1)
- lattice isomorphism theorem (1)
- leading 1 (1)
- leading 1 method (18)
- least common multiple (1)
- left coset (1)
- left ideal (1)
- left module (1)
- Leibniz formula (1)
- length (9)
- length of a vector (13)
- Lie algebra (2)
- limit (4)
- line (4)
- linear algebra (477)
- linear combination (62)
- linear dependent (4)
- linear dynamical system (2)
- linear equation (6)
- linear equations (1)
- linear function (1)
- linear independent (15)
- linearity (3)
- linearity of expectation (1)
- linearly dependent (18)
- linearly independent (47)
- linearly independent vectors (1)
- linear recurrence relation (5)
- linear system (2)
- linear transformation (74)
- link (1)
- localization (1)
- local ring (1)
- log function (1)
- lower bound (1)
- lower central series (2)
- lung cancer (1)
M
- Machin (1)
- magic square (1)
- magnitude (2)
- magnitude of a vector (4)
- Markov (1)
- Markov’s inequality (1)
- Markov matrix (2)
- math-magic (1)
- matrix (139)
- matrix components (1)
- matrix entry (1)
- matrix equation (3)
- matrix exponential (1)
- matrix exponential of a diagonal matrix (1)
- matrix for a linear transf (1)
- matrix for a linear transformation (11)
- matrix for linear transformation (10)
- matrix multiplication (10)
- matrix operation (5)
- matrix operations (2)
- matrix power (1)
- matrix product (13)
- matrix representation (27)
- matrix representation of a linear transformation (1)
- maximal ideal (15)
- maximal real subfield (1)
- Mersenne number (2)
- Mersenne prime (3)
- Michigan State (2)
- Michigan State.GP (2)
- minimal polynomial (8)
- minor matrix (3)
- MIT (3)
- MIT.LA (3)
- modular arithmetic (2)
- module (12)
- module homomorphism (3)
- module theory (12)
- monic (1)
- monic homomorphism (1)
- monic polynomial (1)
- multiplicative group (7)
N
- Nagoya (4)
- Nagoya.LA (4)
- network (1)
- nilpotent (6)
- nilpotent element (5)
- nilpotent group (2)
- nilpotent ideal (1)
- nilpotent matrix (11)
- nilradical (2)
- Noetherian (2)
- Noetherian ring (2)
- non-negative definite (1)
- non-negative random variable (1)
- non-unit (1)
- nonabelian group (10)
- noncommutative ring (2)
- nonsingular (13)
- nonsingular matrix (52)
- nonzero solution (2)
- nonzero vector (1)
- norm (10)
- normal distribution (1)
- normalizer (4)
- normal matrix (2)
- normal random variable (1)
- normal series (1)
- normal subgroup (48)
- normal Sylow subgroup (11)
- norm of a vector (3)
- Northshield (1)
- no solution (2)
- nullity (25)
- nullity of a linear transformation (5)
- nullity of a matrix (5)
- nullspace (6)
- null space (37)
- null space of a linear transformation (4)
- nullspace of a linear transformation (1)
- null space of a matrix (1)
- nullspace of a matrix (2)
- number of elements (1)
- number theory (5)
O
- odd perfect number (1)
- Ohio State (84)
- Ohio State.LA (86)
- one-line proof (1)
- one-to-one linear transformation (1)
- orbit (1)
- orbit-stabilizer theorem (3)
- order (7)
- order of a group (12)
- order of a ring (1)
- order of group (5)
- orthogonal (3)
- orthogonal basis (6)
- orthogonal complement (2)
- orthogonal matrix (12)
- orthogonal set (2)
- orthogonal transformation (2)
- orthogonal vector (5)
- orthogonal vectors (3)
- orthonormal (1)
- orthonormal basis (8)
- orthonormal set (2)
- orthonormal vector (2)
- outcome (2)
P
- p-group (6)
- parameter (1)
- perfect number (2)
- permutation representatiion (1)
- permutation representation (5)
- perpendicular (4)
- perpendicular vector (3)
- perturbation (1)
- pi (2)
- PID (5)
- Pi day (1)
- pie (1)
- plane (2)
- polynomial (14)
- polynomial ring (4)
- popular problems (1)
- positive definite (5)
- positive definite matrix (4)
- positive eigenvalue (3)
- positive semi-definite matrix (3)
- power of a matrix (7)
- preimage (4)
- primary ideal (1)
- prime (2)
- prime element (1)
- prime ideal (24)
- prime number (6)
- Princeton (2)
- Princeton.LA (2)
- principal ideal (5)
- principal ideal domain (5)
- probability (30)
- probability density function (1)
- probability mass function (1)
- probability matrix (3)
- product group (1)
- product of groups (4)
- product of ideals (2)
- projection (1)
- projective matrix (1)
- pullback (1)
- Purdue (4)
- Purdue.GP (1)
- Purdue.LA (3)
- Pythagorean identity (1)
- Pythagorean triple (1)
Q
- quadratic extension (1)
- quadratic field (5)
- quadratic formula (5)
- quadratic function (2)
- quadratic integer ring (2)
- qualifying exam (8)
- quality test (1)
- quartic field extension (1)
- quiz (18)
- quotient (2)
- quotient field (1)
- quotient group (12)
- quotient ring (8)
R
- radical (1)
- radical ideal (2)
- random variable (2)
- range (19)
- range of a linear transformation (3)
- range of a matarix (1)
- range of a matrix (7)
- rank (27)
- rank-nullity theorem (23)
- rank of a linear transformation (5)
- rank of a matrix (22)
- rank of a system (1)
- rational number (2)
- rational numbers (1)
- real eigenvalue (5)
- real matrix (4)
- real numbers (2)
- recurrence relation (3)
- recursive relationship (1)
- reduced echelon form (4)
- reduced row echelon (1)
- reduced row echelon form (23)
- reduced row echelon form matrix (2)
- reducible ideal (1)
- reflection (2)
- relation (3)
- relatively prime (1)
- remainder (2)
- representation (1)
- representative (1)
- restriction of a linear transformation (1)
- Riemann zeta function (1)
- right ideal (1)
- ring (46)
- ring homomorphism (18)
- ring isomorphism (2)
- ring of Gaussian integers (2)
- ring of integers (3)
- ring theory (61)
- ring with unity (4)
- root of unity (10)
- rotation (2)
- rotation matrix (3)
- row echelon form (4)
- row equivalent (6)
- row rank (1)
- row space (6)
- row space method (4)
- row vector (3)
- rule of Sarrus (3)
S
- sample space (1)
- scalar (4)
- scalar multiplication (3)
- second isomorphism theorem (1)
- semidirect product (2)
- separable (1)
- separable polynomial (2)
- sequence (3)
- series of events (1)
- Sherman-Woodberry formula (2)
- short exact sequence (1)
- sign homomorphism (1)
- similar (5)
- similarity transformation (1)
- similar matrix (7)
- simple group (5)
- simple module (2)
- sin (1)
- sine (1)
- sine function (4)
- sin function (1)
- singular (7)
- singular matrix (24)
- size of a matrix (1)
- skew-symmetric matrix (10)
- slope (1)
- smartphone (1)
- smoker (1)
- smooth function (1)
- solution (1)
- solution set (1)
- solvable group (3)
- span (22)
- spanning set (23)
- special linear group (1)
- splitting field (6)
- square-free (1)
- square matrix (1)
- square root (1)
- square root matrix (4)
- square root of a matrix (3)
- stabilizer (2)
- standard basis (2)
- standard deviation (4)
- standard normal (1)
- Stanford (7)
- Stanford.LA (9)
- sto (1)
- stochastic matrix (2)
- subgroup (40)
- subgroup criteria (2)
- submodule (6)
- submodule criteria (2)
- subnormal series (2)
- subring (2)
- subset (1)
- subspace (94)
- subspace criteria (21)
- substitution matrix (1)
- summary (1)
- sum of cubes (2)
- sum of divisors function (1)
- sum of matrices (1)
- sum of squares (3)
- sum of subspaces (2)
- surjective (9)
- surjective homomorphism (7)
- surjective ring homomorphism (2)
- Sylow’s theorem (19)
- Sylow subgroup (11)
- symmetric group (5)
- symmetric matrix (28)
- system (4)
- system of differential equations (1)
- system of linear equations (39)
T
- Taylor polynomial (1)
- the number of subgroups (1)
- third isomorphism theorem (3)
- torsion (1)
- torsion-free group (1)
- torsion element (2)
- torsion module (1)
- torsion subgroup (1)
- torsion submodule (1)
- total probability (2)
- total probability theorem (1)
- trace (10)
- trace of a matrix (19)
- transition matrix (2)
- transitive (1)
- transitive action (1)
- transpose (29)
- transpose matrix (14)
- transpose of a matrix (4)
- transpose of a vector (2)
- transposition (1)
- triangle inequality (1)
- triangularizable matrix (2)
- triangularization (1)
- triangular matrix (4)
- trick (1)
- trigonometric function (6)
- trigonometry (3)
- trigonometry function (4)
- trigonometry identity (2)
- trivial group (3)
- true or false (10)
- twin prime (1)
- two-headed coin (1)
U
- U.Tokyo (3)
- U.Tokyo.LA (3)
- UFD (2)
- uniform distribution (1)
- union (3)
- union of subgroups (1)
- union of submodules (2)
- union of subspaces (1)
- Unique Factorization Domain (2)
- unique solution (3)
- unit (7)
- unitary matrix (3)
- unit element (3)
- unit vector (3)
- unsolved problem (1)
- upper bound (2)
- upper triangular matrix (13)
V
- Vandermonde determinant (3)
- Vandermonde matrix (4)
- variance (7)
- vector (58)
- vector form for the general solution (4)
- vectors (1)
- vector space (136)
- vector space axioms (1)
- vector space of functions (5)
- vector space of matrices (3)
- vector space of polynomials (9)
- vector space of real functions (1)
- vector space over a finite field (1)
- Vieta (1)
W
- Wagstaff number (1)
- Wagstaff prime (1)
- Wallis (1)
- well-defined (2)
- Wronskian (1)
Y
- y-intersept (1)
Z
- Z-table (1)
- zero divisor (7)
- zero ideal (1)
- zero matrix (6)
- zero polynomial (1)
- zero solution (2)
- zero transformation (1)
- zero vector (10)
- zero vector space (1)
2
- 2 by 2 matrix (1)
- 2-digit number (1)
- 2-sided ideal (1)
- 2017 (1)
- 2018 (1)
