According to the definition of orthonormal set : “two vectors u and v in $R^n$ are said to be orthonormal if they are orthogonal and have length 1, and a set of vectors is said to be an orthonormal set if every vector in the set has length 1 and each pair of distinct vector is orthogonal.”
My question: Is there an equivalence relation on every orthonormal set? I know orthonormal set is symmetric and transivite because:
- For any two vectors v and u in the orthonormal set S, if the dot v • u = 0 then u • v = 0
2.For any vectors u v and w in S, if u • v = 0 and v • w = 0, then u • w is also equal to 0
But is the set reflexive? Orthogonality can only be applied when there are two vectors, does that mean even though v • v $\neq$ 0, there is only one vector(the antecedent is false), the conditional proposition still holds?
Note that for an equivalence relation you must verify $x=x$ but the only vector orthogonal to itself is the zero vector. Thus orthogonality cannot be an equivalence relation.