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The method of images (or method of mirror images) is a mathematical tool for solving differential equations, in which boundary conditions are satisfied by combining a solution not restricted by the boundary conditions with its possibly weighted mirror image. Generally, original singularities are inside the domain of interest but the function is ...
Mirrors and Reflections is aimed at undergraduate mathematics students, and uses an intuitive and heavily visual approach suitable for that level. [1] [2] [3] its readers are expected to already have a solid background in linear algebra and some group theory.
Householder transformations are widely used in numerical linear algebra, for example, to annihilate the entries below the main diagonal of a matrix, [2] to perform QR decompositions and in the first step of the QR algorithm. They are also widely used for transforming to a Hessenberg form.
In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane. Some mathematicians use "flip" as a synonym for "reflection". [2 ...
The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1 and is therefore a constructible number. In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length | | can be constructed with compass and straightedge in a finite number of steps.
Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions. In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis ) of an orthogonal transformation is an orthogonal matrix .