Definition 6.2.1
Let V and W be two vector spaces and let
be a linear mapping. The kernel of
,
denoted
is the subset of V formed of the vectors whose image is the zero vector in W.

.

Proposition 6.2.2
is a vector subspace of V.

Definition 6.2.3
Let V and W be two vector spaces and let
be a linear mapping. The image of
,
denoted
is the subset of W formed of the vectors having a preimage in V.

Proposition 6.2.4
is a vector subspace of W.

Theorem 6.2.5
Let V and W be two vector spaces of finite dimension and let
be a linear mapping.
Then:

Example 6.2.6
Consider the following vector spaces over
: