Definition of kernel of linear transformation

Definition: The kernel (or null space) of a linear transformation (T: V \to W) is the set of all vectors in the domain that map to the zero vector:
[\ker(T) = {\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}}]

The kernel is always a subspace of the domain (V).

Example: For (T: \mathbb{R}^3 \to \mathbb{R}^2) defined by (T(x,y,z) = (x,y)), the kernel is ({(0,0,z) : z \in \mathbb{R}}) (the z-axis).

Example: For (T: \mathbb{R}^2 \to \mathbb{R}^2) defined by (T(x,y) = (x+y, x+y)), the kernel is ({(x,-x) : x \in \mathbb{R}}) (the line (y = -x)).

Key facts:

• (T) is one-to-one iff (\ker(T) = {\mathbf{0}})
• (\dim(\ker T)) is called the nullity of (T)
• The kernel measures how far (T) is from being injective