Orthonormal and Orthogonal Set

Topics: Algebra - Inner Product

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(definition)

Let $S=\{v_1,\dots ,v_n\}$ be a subset of ${\mathbb{R}}^n$. Then, $S$ is an orthonormal set if:

1. $⟨v_i,v_j⟩=0$, $i\ne j$
2. $⟨v_i,v_i⟩=1$

If only (1) is satisfied, then $S$ is called an orthogonal set.

(theorem)

If $S=\{v_1,\dots ,v_n\}$ is an orthogonal non-zero vector set, then $S$ is a linearly independent set.