To find the kernel of the linear transformation T, we need to solve the equation T(x, y, z) = (0, 0, 0).
From the definition of T, we have:
T(x, y, z) = (0, 0, 0) if and only if
(0, 0, 0) = (0x + 0y + 0z, 0x + 0y + 0z, 0x + 0y + 0z)
This means that any vector (x, y, z) in R3 that satisfies 0x + 0y + 0z = 0 is in the kernel of T.
In other words, the kernel of T consists of all vectors of the form (x, y, z) where x, y, and z are any real numbers, since any such vector satisfies the equation 0x + 0y + 0z = 0.
Therefore, the kernel of T is the set of all vectors of the form (x, y, z) where x, y, and z are real numbers, which can be written as:
ker(T) = x, y, z ∈ ℝ = ℝ3.
So, the kernel of T is all real numbers (REALS).