A kernel, also known as null space, is the set of all vectors in the domain of a linear transformation that map to the zero vector in the codomain. In other words, it is the set of all inputs that result in an output of zero.
The kernel and the range of a linear transformation are closely related. The dimension of the kernel is equal to the dimension of the domain minus the dimension of the range. In other words, the dimension of the kernel and the range add up to the dimension of the domain.
Yes, the kernel of a linear transformation can be empty if the only vector that maps to the zero vector is the zero vector itself. This means that the linear transformation is one-to-one, also known as injective.
The kernel of a linear transformation can be calculated by finding the null space of the transformation's corresponding matrix. This can be done using techniques such as Gaussian elimination or row reduction.
The kernel is an important concept in linear algebra as it helps us understand the behavior of linear transformations. It can also be used to determine if a linear transformation is invertible, as a linear transformation is invertible if and only if its kernel only contains the zero vector.