Orthogonality

Orthogonality is the relation between two elements of an inner product space whose inner product is zero. It generalizes perpendicularity from Euclidean geometry to vectors, functions, and other spaces.

Inner product space: A vector space equipped with an inner product, a rule that assigns scalar products to pairs of vectors. Orthogonality is defined by the inner product that equips this space.
Orthogonal complement: The subspace of vectors orthogonal to every vector in a given subspace. It collects all directions perpendicular to a chosen subspace.
Discrete Fourier transform: A finite transform that expresses a sequence as a combination of discrete complex-frequency components. Its frequency vectors are orthogonal, enabling exact coefficient recovery.
Perpendicularity: The geometric relation between lines or directions meeting at a right angle. It is the familiar geometric case generalized by inner-product orthogonality.
Inner product: A function of two vectors that generalizes the dot product and supports notions of length and angle. A pair is orthogonal precisely when this function evaluates to zero.
Orthonormal basis: A basis whose vectors each have unit length and are pairwise orthogonal. Coordinates in this basis can be computed by inner products with its vectors.
Wavelet: A localized oscillating function used to represent signals across different scales and positions. Orthogonal wavelet systems provide nonredundant signal representations.
Linear independence: A property of vectors where no vector in a set is a linear combination of the others. Orthogonality of nonzero vectors guarantees independence, but independence does not guarantee orthogonality.
Norm (mathematics): A function assigning each vector a nonnegative length that satisfies homogeneity and the triangle inequality. The inner product induces lengths, letting orthogonality yield the Pythagorean identity.
Gram–Schmidt process: An algorithm that converts a linearly independent list of vectors into an orthonormal list spanning the same subspace. It constructs orthogonal directions from vectors that are not initially perpendicular.

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