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The 17 pages that link to Orthonormal basis, each with the reason it gives.

• Hilbert spaceRelated: Coordinates along an orthonormal basis represent vectors and preserve their inner products.

• Euclidean spaceRelated: Such a basis turns the inner product into the familiar sum-of-products formula.

• Rotation matrixRelated: The columns of a rotation matrix form an orthonormal basis.

• Orthogonal projectionRelated: Coordinates along these vectors make projection especially simple.

• Euclidean normRelated: In orthonormal coordinates, the Euclidean norm is the square root of the sum of coordinate squares.

• Gram matrixRelated: Coordinates in an orthonormal basis preserve inner products and therefore the Gram matrix.

• QR decompositionRelated: The columns of Q are orthonormal, spanning the column space in a reduced decomposition.

• Probability amplitudeRelated: Amplitudes are basis-dependent coefficients of a state vector.

• Adjoint operatorRelated: Matrix representations in orthonormal bases make the adjoint a conjugate transpose.

• Kronecker deltaRelated: Its basis-vector inner products are encoded by delta values.

• Parseval's identityRelated: Fourier coefficients are coordinates in an orthonormal system, making squared norms sums of squared coordinates.

• Parseval's theoremRelated: Fourier coefficients in an orthonormal basis contribute independently to squared norm.