Question

Find the kernel of the linear transformation. (If all real numbers are solutions, enter REALS.) T: R3 → R3, T(x, y, z) = (0, 0, 0) ker(T) = * :*,VZER : x, Y, ZE x Need Help? Read It Submit Answer

Answer 1

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).