every orthonormal set is linearly independent
Proof. We denote by the inner product of . Let be an orthonormal set of vectors. Let us first consider the case when is finite, i.e., for some . Suppose
for some scalars (belonging to the field on the underlying vector space of ). For a fixed in , we then have
so , and is linearly independent. Next, suppose is infinite (countable or uncountable). To prove that is linearly independent, we need to show that all finite subsets of 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|
|Date of creation||2013-03-22 13:33:48|
|Last modified on||2013-03-22 13:33:48|
|Last modified by||mathcam (2727)|