Basis22

Basis22 refers to a named basis in a 22-dimensional vector space, typically denoted by B = {b1, b2, ..., b22}, where the vectors are linearly independent and span the space V over a field F (usually the real numbers R or complex numbers C). The basis provides a coordinate system for V: every vector v ∈ V has a unique coordinate vector [v] = (v1, ..., v22) such that v = Σ vi bi. The change of basis between B and the standard basis is represented by the 22×22 matrix whose columns are the basis vectors, and this matrix provides the coordinate transformation between representations.

If the basis is orthonormal, B is an orthonormal basis and the transformation matrix is orthogonal (in R) or unitary (in C), preserving inner products.

Construction: Basis22 can be formed by selecting 22 linearly independent vectors in V. It can also be produced by applying Gram-Schmidt to any linearly independent set of 22 vectors, or by applying a linear transformation to the standard basis. The most common simple instance in R^22 is the standard basis e1,...,e22, which itself constitutes a basis22 for that space.

Applications: A basis22 provides a fixed frame for high-dimensional computations, including projections, coordinate transforms, and linear algorithms in 22 dimensions, as well as representations of states or features in applications such as machine learning, physics, or numerical linear algebra.

See also: basis, vector space, orthonormal basis, coordinate transformation, Gram-Schmidt, linear independence.