Orthonormal Basis - Examples

Examples

  • The set of vectors {e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1)} (the standard basis) forms an orthonormal basis of R3.
Proof: A straightforward computation shows that the inner products of these vectors equals zero, <e1, e2> = <e1, e3> = <e2, e3> = 0 and that each of their magnitudes equals one, ||e1|| = ||e2|| = ||e3|| = 1. This means {e1, e2, e3} is an orthonormal set. All vectors (x, y, z) in R3 can be expressed as a sum of the basis vectors scaled
so {e1,e2,e3} spans R3 and hence must be a basis. It may also be shown that the standard basis rotated about an axis through the origin or reflected in a plane through the origin forms an orthonormal basis of R3.
  • The set {fn : n ∈ Z} with fn(x) = exp(2πinx) forms an orthonormal basis of the space of functions with finite Lebesgue integrals, L2, with respect to the 2 norm. This is fundamental to the study of Fourier series.
  • The set {eb : b ∈ B} with eb(c) = 1 if b = c and 0 otherwise forms an orthonormal basis of ℓ 2(B).
  • Eigenfunctions of a Sturm–Liouville eigenproblem.
  • An orthogonal matrix is a matrix whose column vectors form an orthonormal set.

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