every orthonormal set is linearly independent

Proof. We denote by , the inner productMathworldPlanetmath of L. Let S be an orthonormal set of vectors. Let us first consider the case when S is finite, i.e., S={e1,,en} for some n. Suppose


for some scalars λi (belonging to the field on the underlying vector spaceMathworldPlanetmath of L). For a fixed k in 1,,n, we then have


so λk=0, and S is linearly independent. Next, suppose S is infiniteMathworldPlanetmath (countableMathworldPlanetmath or uncountable). To prove that S is linearly independent, we need to show that all finite subsets of S are linearly independent. Since any subset of an orthonormal set is also orthonormal, the infinite case follows from the finite case.

Title every orthonormal set is linearly independent
Canonical name EveryOrthonormalSetIsLinearlyIndependent
Date of creation 2013-03-22 13:33:48
Last modified on 2013-03-22 13:33:48
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Theorem
Classification msc 15A63