The Rotation Problem and Hamilton's Discovery of Quaternions - Part 4
Practical implementation of quaternions for 3D rotations, focusing on half-turns and vector trigonometry. Explores quaternion multiplication and its geometric interpretation for rotations.
The Rotation Problem and Hamilton's Discovery of Quaternions - Part 2 - Famous Math Problems 13b
Explore 3D rotations, their composition, and algebraic representation. Learn about Euler's theorem, vector operations, and the foundations for understanding quaternions in graphics and rocket science.
More Homology Computations in Algebraic Topology - Lecture 2
Explore homology computations for the 2-sphere using tetrahedron and delta-complex models. Learn how linear algebra applies to topological problems and simplify calculations with semi-simplicial complexes.
Building On-Sequences: From Constant to Polynumber - Math Foundations 100
Explores the concept of on-sequences, starting with constant and polynumber types. Covers arithmetic operations, visualization techniques, and connections to the Online Encyclopedia of Integer Sequences.
Commutative Groups: Isomorphisms, Homomorphisms, Cosets and Quotient Groups - Lecture 3
Explore commutative groups, isomorphisms, homomorphisms, cosets, and quotient groups. Learn the fundamental structure theorem and its applications in understanding group theory concepts.
Introduction to Group Theory
Brief overview of group theory focusing on commutative groups, covering key concepts, examples, and the Fundamental Theorem of Finite Commutative Groups.
Colored Spreads and Generalizations in Rational Trigonometry - Lecture 2
Explore colored spreads in planar geometries, visualizing red and green spreads using cross products and determinants. Learn about generalized dot products and Pythagoras' theorem in different geometries.
Coloured Spreads and Generalizations in Rational Trigonometry - Lecture 1
Explore colored spreads theorem, planar dot products, and geometric interpretations of quadrances in blue, red, and green geometries. Gain insights into rational trigonometry and linear algebra concepts.
Relations Between Multisets - Abstract Algebra Math Foundations - Lecture 225
Explore integral linear spans and relations between msets, drawing parallels to linear algebra's rank-nullity theorem. Learn to obtain relations using the Hermite normal form algorithm.
Integral Row Reduction and Hermite Normal Form - Math Foundations 223
Explores integral row reduction algorithm for finding a basis of integral linear space, paralleling Euclid's algorithm. Introduces Hermite normal form for msets of vexels in abstract algebra context.
Bases and Dimension for Integral Linear Spaces - Math Foundations 222
Explores algorithm for finding simplified bases of integral linear spaces, adapting Gaussian elimination to work with integral combinations. Introduces key concepts in abstract algebra through practical computation.
Lattice Relations and Hermite Normal Form - Abstract Algebra Math Foundations 224
Explore lattices and integral linear spans of vexels, including the Hermite normal form algorithm for understanding lattice structure and relations between elements in a spanning set.
Linear Spaces and Spans II - MathFoundations 220
Introduction to integral linear spaces as abstract algebraic objects, exploring their applications in mathematics, chemistry, and physics. Foundational concepts for linear algebra using vexel language for concrete examples.
Linear Spaces and Spans I - Abstract Algebra Math Foundations 219
Introduction to linear spaces and spans, generalizing natural numbers with addition to form spans of multisets. Explores fundamental concepts in abstract algebra and linear algebra.
Rethinking Roots of Unity in Complex Algebra - Math Foundations 218
Explores complex roots of unity, challenging traditional views and highlighting logical difficulties in their definition and existence, emphasizing rational parametrization and algebraic approaches over infinite processes.
Fundamental Theorem of Algebra Reexamined - Math Foundations 217
Reexamines the Fundamental Theorem of Algebra, proposing new formulations based on rational complex arithmetic and quadrance, avoiding assumptions about continuity and real/complex numbers.
The Fundamental Dream of Algebra - Math Foundations 216
Challenging the Fundamental Theorem of Algebra, this video explores its flaws and proposes new thinking in mathematics, connecting to broader issues in the field.
Logical Challenges in Abstract Algebra - Math Foundations 214
Explores logical difficulties in abstract algebra, focusing on defining groups and the impact of different definitions on group theory development. Emphasizes critical thinking in mathematical foundations.
Algebra with Maxels from Integers - Data Structures Math Foundations 209
Exploring maxel theory as an extension of matrix theory, introducing shift invariance and partial identity maxels to restructure linear algebra with integer indices.
Integral Vectors and Matrices via Vexels and Maxels - Data Structures Math Foundations
Explore integral vectors and matrices through vexels and maxels, expanding on previous concepts with a multiset framework for more flexible indexing and generalized representations.
Reciprocals, Powers of 10, and Euler's Totient Function - Math Foundations 202
Explore decimal expansions of fractions, their connection to modular arithmetic, and Euler's totient function. Gain insights into repeated decimal cycles and their relationship with powers of 10 mod n.