Definition:Orthogonal Basis

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Definition

Orthogonal Basis of Vector Space

Let $V$ be a vector space.

Let $\BB = \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be a basis of $V$.


Then $\BB$ is an orthogonal basis if and only if $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ are pairwise perpendicular.


Orthogonal Basis of Inner Product Space

Definition:Orthogonal Basis of Inner Product Space

Orthogonal Basis of Bilinear Space

Let $\mathbb K$ be a field.

Let $\struct {V, f}$ be a bilinear space over $\mathbb K$ of finite dimension $n > 0$.

Let $\BB = \tuple {b_1, \ldots, b_n}$ be an ordered basis of $V$.


Then $\BB$ is orthogonal if and only if:

$\map f {b_i, b_j} = 0$

for $i \ne j$.


That is, if and only if the matrix of $f$ relative to $\BB$ is diagonal.