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  2. Orthogonality (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Orthogonality_(mathematics)

    In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B {\displaystyle B} are orthogonal when B ( u , v ) = 0 {\displaystyle B(\mathbf {u} ,\mathbf {v} )=0} .

  3. Orthogonality - Wikipedia

    en.wikipedia.org/wiki/Orthogonality

    The line segments AB and CD are orthogonal to each other. In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.Whereas perpendicular is typically followed by to when relating two lines to one another (e.g., "line A is perpendicular to line B"), [1] orthogonal is commonly used without to (e.g., "orthogonal lines A and B").

  4. Orthogonal functions - Wikipedia

    en.wikipedia.org/wiki/Orthogonal_functions

    In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: , = ¯ ().

  5. Orthogonal matrix - Wikipedia

    en.wikipedia.org/wiki/Orthogonal_matrix

    Orthogonal matrices are important for a number of reasons, both theoretical and practical. The n × n orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by O(n), which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the point group of a

  6. Gram–Schmidt process - Wikipedia

    en.wikipedia.org/wiki/Gram–Schmidt_process

    The first two steps of the Gram–Schmidt process. In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other.

  7. Orthonormality - Wikipedia

    en.wikipedia.org/wiki/Orthonormality

    The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle).

  8. Orthogonal transformation - Wikipedia

    en.wikipedia.org/wiki/Orthogonal_transformation

    In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product.That is, for each pair u, v of elements of V, we have [1]

  9. Orthogonalization - Wikipedia

    en.wikipedia.org/wiki/Orthogonalization

    In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.Formally, starting with a linearly independent set of vectors {v 1, ... , v k} in an inner product space (most commonly the Euclidean space R n), orthogonalization results in a set of orthogonal vectors {u 1, ... , u k} that generate the same subspace as the vectors v 1 ...