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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 ...
The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form .Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form , = ((+) ()) allows vectors and to be defined as being orthogonal with respect to when (+) () = .
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In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° (radians), or one of the vectors is zero. [4] Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension.
A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length.
be the orthogonal projection onto the normal vector at a, so that = is the orthogonal projection onto the tangent space at a. The group G = SO(3) acts by rotation on E 3 leaving S 2 invariant. The stabilizer subgroup K of the vector (1,0,0) in E 3 may be identified with SO(2) and hence S 2 may be identified with SO(3)/SO(2).
Plot of the Chebyshev rational functions of order n=0,1,2,3 and 4 between x=0.01 and 100. Legendre and Chebyshev polynomials provide orthogonal families for the interval [−1, 1] while occasionally orthogonal families are required on [0, ∞). In this case it is convenient to apply the Cayley transform first, to bring the argument into [−1, 1].
A lamellar vector field is a special case given by vector fields with zero curl. The adjective "lamellar" derives from the noun "lamella", which means a thin layer. The lamellae to which "lamellar vector field" refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field. [1]