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In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. 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.
Query-Key normalization (QKNorm) [32] normalizes query and key vectors to have unit L2 norm. In nGPT, many vectors are normalized to have unit L2 norm: [33] hidden state vectors, input and output embedding vectors, weight matrix columns, and query and key vectors.
A polygon and its two normal vectors A normal to a surface at a point is the same as a normal to the tangent plane to the surface at the same point.. In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object.
[1] [2] The term normalized vector is sometimes used as a synonym for unit vector. A unit vector is often used to represent directions, such as normal directions. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination form of unit vectors.
For a general inner product space , an orthonormal basis can be used to define normalized orthogonal coordinates on . Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of R n {\displaystyle \mathbb {R ...
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.
A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. [1] A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings.
However, normal may also refer to the magnitude of a vector. 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.