Find a normal vector to the hyperplane in R4 spanned by a. (1, 1, 1, 1), (1, 2, 1, 2)

Step 1 of  4

a) We want to find a normal vector to the hyperplane in \(R^{4}\) spanned by the vectors \(v_{1}=(1,1,1,1), v_{2}=(1,2,1,2)\) and \(v_{3}=(1,3,2,4)\) that is we want a vector \(x \in \mid R^{4}\) satisfying the system of equations.

\(v_{1} x=v_{2} x=v_{3} x=0\) such a vector \(x\) must satisfy the system of equations

\(\begin{array}{l} x_{1}+x_{2}+x_{3}+x_{4}=0 \\ x_{1}+2 x_{2}+x_{3}+2 x_{4}=0 \\ x_{1}+3 x_{2}+2 x_{3}+4 x_{4}=0 \end{array}\)