orthonormal

+Orthonormal

+In linear algebra and functional analysis, the term orthonormal refers to a set of vectors that are both orthogonal and normalized. This property is significant in various applications including signal processing, computer graphics, and quantum mechanics.

+Definition

+A set of vectors {v₁, v₂, ..., v_n} in an inner product space is orthonormal if the following conditions are satisfied:

• +Each vector is of unit length: ||v_i|| = 1 for all i
• +The vectors are mutually perpendicular: ⟨v_i, v_j⟩ = 0 for all i ≠ j

+Properties

• +Any linear combination of orthonormal vectors remains orthogonal if the coefficients are appropriately scaled.
• +If a set of vectors is orthonormal, it forms a basis for a vector space, allowing any vector in that space to be represented uniquely as a linear combination.

+Applications

Signal Processing

+Orthonormal bases are used in Fourier transforms and wavelet transforms for efficient data representation.

Computer Graphics

+Orthonormal vectors are essential for defining coordinate systems and transformations in 3D graphics.

Quantum Mechanics

+States of a quantum system are often represented by orthonormal vectors in Hilbert space.