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His channel has 4.2 million subscribers in YouTube, and more than 500 million views [4] [5] Known as Julioprofe, he appeared in e-learning videos of free education in the areas of: algebra, geometry, trigonometry, analytic geometry, computing, physics, linear algebra and higher mathematics.
Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional vector space. For a more general coefficient ring, the same construction works if we take the free module on n generators. The construction of the free algebra on E is functorial in nature and satisfies an appropriate universal property.
He retired on May 15, 2023 after giving his final Linear Algebra and Learning from Data [6] lecture at MIT. [7] Strang's teaching has focused on linear algebra which has helped the subject become essential for students of many majors. His linear algebra video lectures are popular on YouTube and MIT OpenCourseware.
Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic geometry. This core functionality includes arithmetic on rings, modules, and matrices, as well as algorithms for Gröbner bases, free resolutions, Hilbert series, determinants and Pfaffians, factoring, and similar.
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
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Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module. If M n ( R ) is the ring of n × n matrices over a ring R , M is an M n ( R )-module, and e i is the n × n matrix with 1 in the ( i , i ) -entry (and zeros elsewhere), then e i M is an ...
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