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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. An orthonormal set which forms a basis is called an orthonormal basis.
A variant of the Gram–Schmidt process using transfinite recursion applied to a (possibly uncountably) infinite sequence of vectors () < yields a set of orthonormal vectors () < with such that for any , the completion of the span of {: < (,)} is the same as that of {: <}.
A set of mutually orthonormal vectors in a Hilbert space is called an orthonormal system. An orthonormal basis is an orthonormal system with the additional property that the linear span of S {\displaystyle S} is dense in H {\displaystyle H} . [ 6 ]
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q T Q = Q Q T = I , {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,} where Q T is the transpose of Q and I is the identity matrix .
In general, any operator in a Hilbert space that acts by permuting an orthonormal basis is unitary. In the finite dimensional case, such operators are the permutation matrices. On the vector space C of complex numbers, multiplication by a number of absolute value 1, that is, a number of the form e iθ for θ ∈ R, is a unitary operator.
The fiber over a given point P in () is the set of all orthonormal k-frames contained in the space P. This projection has the structure of a principal G-bundle where G is the associated classical group of degree k. Take the real case for concreteness.
Move over, Wordle, Connections and Mini Crossword—there's a new NYT word game in town! The New York Times' recent game, "Strands," is becoming more and more popular as another daily activity ...
In particular, a set is called orthonormal (orthogonal plus normal) if it is an orthogonal set of unit vectors. As a result, use of the term normal to mean "orthogonal" is often avoided. The word "normal" also has a different meaning in probability and statistics. A vector space with a bilinear form generalizes the case of an inner product.