Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d].
-$\left[ \begin{array} { l } 1 \ 1 \ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \ - 1 \ 5 \end{array} \right] , \left[ \begin{array} { r } 4 \ 2 \ - 2 \end{array} \right]$
A) $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = x _ { 1 } + x _ { 2 } + \frac { 5 } { 2 } x _ { 3 } , d = \frac { 9 } { 2 }$
B) $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = - 2 x _ { 1 } + 9 x _ { 2 } + x _ { 3 } , d = 8$
C) $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = 8 x _ { 1 } + 2 x _ { 2 } + 4 x _ { 3 } , d = 14$
D) $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = x _ { 1 } + 9 x _ { 2 } + 4 x _ { 3 } , d = 14$

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The Answer of Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d].
-$\left[ \begin{array} { l } 1 \ 1 \ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \ - 1 \ 5 \end{array} \right] , \left[ \begin{array} { r } 4 \ 2 \ - 2 \end{array} \right]$
A) $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = x _ { 1 } + x _ { 2 } + \frac { 5 } { 2 } x _ { 3 } , d = \frac { 9 } { 2 }$
B) $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = - 2 x _ { 1 } + 9 x _ { 2 } + x _ { 3 } , d = 8$
C) $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = 8 x _ { 1 } + 2 x _ { 2 } + 4 x _ { 3 } , d = 14$
D) $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = x _ { 1 } + 9 x _ { 2 } + 4 x _ { 3 } , d = 14$

Let H Be the Hyperplane Through the Points A) f(x1,x2,x3)=x1+x2+52x3,d=92f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = x _ { 1 } + x _ { 2 } + \frac { 5 } { 2 } x _ { 3 } , d = \frac { 9 } { 2 }f(x1​,x2​,x3​)=x1​+x2​+25​x3​,d=29​

Let H Be the Hyperplane Through the Points A) $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = x _ { 1 } + x _ { 2 } + \frac { 5 } { 2 } x _ { 3 } , d = \frac { 9 } { 2 }$