Kernel and image of a linear transformation

Let and be vector spaces, and be a linear transformation.

Definition ( kernel)

The kernel of a linear transformation is the set of vectors in that map to the vector. In quantifiers,

Definition ( image)

The image of a linear transformation is the set of vectors in that can be obtained as the result of applying the transformation to a vector in . In quantifiers,

Lemma

• is a subspace in .
• is a subspace in .