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
**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