Final answer:
The kernel of the linear transformation T:R² → R², where T(x, y) = (x + 4y, y - x), is the set containing only the zero vector, {(0,0)}, found by solving the system of equations created by setting the transformation equal to the zero vector.
Step-by-step explanation:
The question asks us to find the kernel of the linear transformation T:R² → R², where T(x, y) = (x + 4y, y - x). The kernel of a linear transformation consists of all vectors in the domain that map to the zero vector in the codomain. To find the kernel, we need to solve the equation T(x, y) = (0, 0). This results in a system of equations:
Solving this system of equations, we set the equations equal to zero:
- x + 4y = 0 → x = -4y
- y - x = 0 → y = x
Since y = x, substituting into the first equation gives x + 4(x) = 0, which simplifies to 5x = 0. So, x = 0 and consequently,