F(x,y,z)=U(x,y,z) relates the function F to the linear transformation U. F maps a 3D point (x,y,z) to a different point in 3-space.
U can be written as a matrix:
( 5 8 16 )
( 4 1 8 )
( -4 -4 -11 )
When this matrix is applied to matrix:
( x )
( y )
( z )
the given transformation takes place.
I’m not sure what you mean by the root of functions, but I guess you are talking about linear functions of the type A.U + B = 0 where A and B are 3×3 matrices. Matrix A is:
( a₁₁ a₁₂ a₁₃ )
( a₂₁ a₂₂ a₂₃ )
( a₃₁ a₃₂ a₃₃ )
Similarly for matrix B with elements b₁₁, b₁₂, etc., where b₁₁=-(5a₁₁+8a₁₂+16a₂₃), b₁₂=-(8a₁₁+a₁₂-4a₁₃), b₁₃=-(16a₁₁+8a₁₂-11a₁₃), and so on. If these conditions are met, U would be a root of the linear function. I hope this helps.