Tag Archives: algebra

Published on Feb 18, 2016
This problem has over 2 million comments on Facebook and is getting coverage in mainstream media and blogs. Can you figure out this Facebook fruit algebra puzzle involving apples, bananas, and coconuts? The video presents what many people consider to be the correct answer.

Published on Jun 28, 2015
What is the car’s parking spot number? Hong Kong elementary students were asked to solve this problem in 20 seconds. The problem stumped a lot of adults and went viral in 2014.

the Archimedean property

In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder.[1]

The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert‘s axioms for geometry, and the theories of ordered groups, ordered fields, and local fields.

An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, a linearly ordered group that is Archimedean is an Archimedean group.

This can be made precise in various contexts with slightly different ways of formulation. For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.

Spectral theorem

Spectral theorem

From Wikipedia, the free encyclopedia
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can bediagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.
Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.
The spectral theorem also provides a canonical decomposition, called the spectral decompositioneigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.
In this article we consider mainly the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.

Linear algebraic group

Linear algebraic group

From Wikipedia, the free encyclopedia
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices
(under matrix multiplication) that is defined by polynomial equations.

An example is the orthogonal group, defined by the relation MTM = I where MT is the transpose of
M.

magmas

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sortsclosed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra.


Types of magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include
Magma to Group.svg

From magma to group, via two alternative paths. Key:
M = magma, d = divisibility, a = associativity,
Q = quasigroup, S = semigroup, e = identity.
L = loop, i = invertibility, N = monoid, G = group
Note that both divisibility and invertibility imply
the existence of the cancellation property.

Fields

 

In abstract algebra, a field is an algebraic structure with notions of additionsubtractionmultiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fieldsp-adic fields, and so forth.
Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomialswith coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry.
As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fields allow for division (though not division by zero), while a ring need not possessmultiplicative inverses. Also, the multiplication operation in a field is required to be commutative. A ring in which division is possible but commutativity is not assumed (such as the quaternions) is called a division ring or skew field. (Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.)
As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following (not exhaustive) chain of class inclusions:

Rotation matrix

Rotation matrix

From Wikipedia, the free encyclopedia
In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix
R =  begin{bmatrix} cos theta & -sin theta \ sin theta & cos theta \ end{bmatrix}
rotates points in the xyCartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details).
In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometryphysics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space.
Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, that is an orthogonal matrix whose determinant is 1:
R^{T} = R^{-1}, det R = 1,.
The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1.
http://en.wikipedia.org/wiki/Rotation_matrix

As in two dimensions a matrix can be used to rotate a point (xyz) to a point (x′, y′, z′). The matrix used is a 3 × 3 matrix,
mathbf{A} = begin{pmatrix} a & b & c \ d & e & f \ g & h & i  end{pmatrix}
This is multiplied by a vector representing the point to give the result
  mathbf{A}  begin{pmatrix} x \ y \ z end{pmatrix} =  begin{pmatrix} a & b & c \ d & e & f \ g & h & i  end{pmatrix}  begin{pmatrix} x \ y \ z end{pmatrix} =  begin{pmatrix} x' \ y' \ z' end{pmatrix}
The matrix A is a member of the three dimensional special orthogonal group, SO(3), that is it is an orthogonal matrix withdeterminant 1. That it is an orthogonal matrix means that its rows are a set of orthogonal unit vectors (so they are an orthonormal basis) as are its columns, making it easy to spot and check if a matrix is a valid rotation matrix. The determinant must be 1 as if it is -1 (the only other possibility for an orthogonal matrix) then the transformation given by it is a reflectionimproper rotation or inversion in a point, i.e. not a rotation.
Matrices are often used for doing transformations, especially when a large number of points are being transformed, as they are a direct representation of the linear operator. Rotations represented in other ways are often converted to matrices before being used. They can be extended to represent rotations and transformations at the same time using Homogeneous coordinates. Transformations in this space are represented by 4 × 4 matrices, which are not rotation matrices but which have a 3 × 3 rotation matrix in the upper left corner.
The main disadvantage of matrices is that they are more expensive to calculate and do calculations with. Also in calculations wherenumerical instability is a concern matrices can be more prone to it, so calculations to restore orthonormality, which are expensive to do for matrices, need to be done more often.