Classical Yang-Baxter Equation and Some Related Algebraic
... equivalent to a linear map r : g∗ → g satisfying [r(x), r(y)] = r(ad∗ r(x)(y) − ad∗ r(y)(x)), ∀x, y ∈ g∗ , ...
... equivalent to a linear map r : g∗ → g satisfying [r(x), r(y)] = r(ad∗ r(x)(y) − ad∗ r(y)(x)), ∀x, y ∈ g∗ , ...
Document
... added. Work backward: Subtract 10 from both sides. Since a is multiplied by 4, divide both sides by 4 to undo the multiplication. ...
... added. Work backward: Subtract 10 from both sides. Since a is multiplied by 4, divide both sides by 4 to undo the multiplication. ...
Groups
... on a set X with more then one element are non-commutative operations. Example 10. Composition of linear operators on the set of all linear operators on any linear (vector) space of the dimension more then one dimX > 1 over any field with more then one element is a non-commutative operation. The matr ...
... on a set X with more then one element are non-commutative operations. Example 10. Composition of linear operators on the set of all linear operators on any linear (vector) space of the dimension more then one dimX > 1 over any field with more then one element is a non-commutative operation. The matr ...
Linear Algebra for Theoretical Neuroscience (Part 2) 4 Complex
... eigenvalues; but once we extend our number system to include complex numbers, every real Ndimensional matrix has N eigenvalues, and more generally every complex N-dimensional matrix has N eigenvalues. (The reason is exactly the same as in analysis: every N-dimensional matrix has an associated Nth or ...
... eigenvalues; but once we extend our number system to include complex numbers, every real Ndimensional matrix has N eigenvalues, and more generally every complex N-dimensional matrix has N eigenvalues. (The reason is exactly the same as in analysis: every N-dimensional matrix has an associated Nth or ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.