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More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and .
These equations, often complex and non-linear, can be linearized using linear algebra methods, allowing for simpler solutions and analyses. In the field of fluid dynamics, linear algebra finds its application in computational fluid dynamics (CFD), a branch that uses numerical analysis and data structures to solve and analyze problems involving ...
The fundamental example of a linear complex structure is the structure on R 2n coming from the complex structure on C n.That is, the complex n-dimensional space C n is also a real 2n-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number i is not only a complex linear transform of the space, thought of as a complex ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
An algebraic structure is a framework for understanding operations on mathematical objects, like the addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what ...
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring.If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and ...
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical ...
For example, the position and the linear momentum in the -direction of a particle are represented by the operators and , respectively (where is the reduced Planck constant). This is the same example except for the constant − i ℏ {\displaystyle -i\hbar } , so again the operators do not commute and the physical meaning is that the position ...
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