U: R³ -> R³ , U(x, y, z) = (5x+8y+16z,4x+y+8z,-4x-4y-11z) is a lineer transformation, find the functions that accept U as a root.

Question

The topics are Linear Operators and Linear Transformations. Please tell me if i made some mistakes while translating the question to english if or there is something that doesn't make sense.

Answer 1

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.