Answer
$\{(-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}),(\frac{2}{\sqrt 6},\frac{1}{\sqrt 6},0,\frac{1}{\sqrt 6})\}$
Work Step by Step
Let $x_1=(-1,1,1,1) \\
x_2=(1,2,1,2)$
According to Gram-Schmidt process, we have:
$v_1=x_1=(-1,1,1,1)\\
v_2=x_2-\frac{}{||v_1||^2}v_1\\
=(1,2,1,2)-\frac{((-1,1,1,1),(1,2,1,2)}{||(-1,1,1,1)||^2} (-1,1,1,1)\\
=(1,2,1,2)-\frac{(-1).1+1.2+1.1+1.2}{1^2+1^2+1^2+(-1)^2} (-1,1,1,1)\\
=(2,1,0,1)$
To determine an orthogonal set, we obtain:
$\frac{v_1}{||v_1||}=\frac{(-1,1,1,1)}{\sqrt (-1)^2+1^2+1^2+1^2}=\frac{(-1,1,1,1)}{\sqrt 4}=(-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$
$\frac{v_2}{||v_2||}=\frac{(2,1,0,1)}{\sqrt (2^2+1^2+0+1^2)}=\frac{(2,1,0,1)}{\sqrt 6}=(\frac{2}{\sqrt 6},\frac{1}{\sqrt 6},0,\frac{1}{\sqrt 6})$
Hence, a corresponding orthonormal set of vector is :
$\{(-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}),(\frac{2}{\sqrt 6},\frac{1}{\sqrt 6},0,\frac{1}{\sqrt 6})\}$
