Final answer:
The kernel of a linear transformation is the set of inputs that map to the zero vector, while the range is the set of all possible output vectors.
Step-by-step explanation:
A linear transformation is a function that maps vectors from one vector space to another, while preserving the linear structure of the spaces. The kernel of a linear transformation is the set of all vectors that map to the zero vector in the codomain. In other words, it is the set of all inputs that are mapped to zero by the transformation. The range of a linear transformation is the set of all possible output values.
To find the kernel of a linear transformation, we need to solve the equation T(x) = 0, where T is the linear transformation and x is a vector. This equation represents the vectors in the domain that map to the zero vector in the codomain. The solutions to this equation form the kernel of the transformation.
To find the range of a linear transformation, we need to determine all possible output values. This can be done by applying the transformation to a basis for the domain space, and then examining the resulting vectors in the codomain. The set of all possible output vectors forms the range of the transformation.