Questão 5: Lembrando que:
Processo de Gram-Schmidt
Para converter uma base
\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{r}\right\} numa base ortogonal
\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{r}\right\}, efetue as seguintes contas.
Passo 1.
\mathbf{v}_{1}=\mathbf{u}_{1}
Passo 2.
\quad \mathbf{v}_{2}=\mathbf{u}_{1}-\frac{\left\langle\mathbf{u}_{2}, \mathbf{v}_{1}\right\rangle}{\left\|\mathbf{v}_{1}\right\|^{2}} \mathbf{v}_{1}
Passo 3.
\quad \mathbf{v}_{3}=\mathbf{u}_{3}-\frac{\left\langle\mathbf{u}_{3}, \mathbf{v}_{1}\right\rangle}{\left\|\mathbf{v}_{1}\right\|^{2}} \mathbf{v}_{1}-\frac{\left\langle\mathbf{u}_{3}, \mathbf{v}_{2}\right\rangle}{\left\|\mathbf{v}_{2}\right\|^{2}} \mathbf{v}_{2}
Passo 4.
\quad \mathbf{v}_{4}=\mathbf{u}_{4}-\frac{\left\langle\mathbf{u}_{4}, \mathbf{v}_{1}\right\rangle}{\left\|\mathbf{v}_{1}\right\|^{2}} \mathbf{v}_{1}-\frac{\left\langle\mathbf{u}_{4}, \mathbf{v}_{2}\right\rangle}{\left\|\mathbf{v}_{2}\right\|^{2}} \mathbf{v}_{2}-\frac{\left\langle\mathbf{u}_{4}, \mathbf{v}_{3}\right\rangle}{\left\|\mathbf{v}_{3}\right\|^{2}} \mathbf{v}_{3}
(continue até
r passos)
Passo opcional. Para converter a base ortogonal numa base ortonormal
\left\{\mathbf{q}_{1}, \mathbf{q}_{2}, \ldots, \mathbf{q}_{r}\right\}, normalize os vetores da base ortogonal.
Usando o processo de Gram-Schmidt transforme os vetores de base
u_{1}=(1,1,1), u_{2}=(0,1,1),
u_{3}=(0,0,1) em uma base ortogonal
\left\{v_{1}, v_{2}, v_{3}\right\} e, depois, normalize os vetores da base ortogonal para obter uma base ortonormal
\left\{q_{1}, q_{2}, q_{3}\right\}.