Para esse exercicio devemos encontrar uma base ortogonal para s e para isso realizaremos os cálculos abaixo:

\(\begin{align} & a=(0,1,-1,0) \\ & b=(1,-1,0,0) \\ & \\ & {{a}_{2}}=(1,-1,0,0)-\frac{1}{||1||}(0,1,-1,0) \\ & {{a}_{2}}=(1,-1,0,0)-(0,1,-1,0) \\ & {{a}_{2}}=(1,-2,1,0) \\ & \\ & {{b}_{2}}=(0,1,-1,0)-\frac{1}{||1||}(1,-1,0,0) \\ & {{b}_{2}}=(0,1,-1,0)-(1,-1,0,0) \\ & {{b}_{2}}=(-1,2,-1,0) \\ & \\ & B=\left\{ \left( 1,-2,1,0 \right),\left( -1,2,-1,0 \right) \right\} \\ \end{align}\ \)

Portanto, a base ortohonal será:

\(\boxed{B = \left\{ {\left( {1, - 2,1,0} \right),\left( { - 1,2, - 1,0} \right)} \right\}}\)

b) A base perpendicular é a mesma do item acima.