Kernel and Range

Overview

The kernel and range are fundamental subspaces associated with a linear transformation. They describe what gets mapped to zero and what can be reached by the transformation.

Kernel (Null Space)

The kernel of $T: V \to W$ is:

$\ker(T) = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}\}$

The set of all vectors that map to zero.

Properties

• $\ker(T)$ is a subspace of $V$
• $\dim(\ker(T))$ = nullity of $T$
• $T$ is injective (one-to-one) $\Leftrightarrow \ker(T) = \{\mathbf{0}\}$

Example

For $T(x, y, z) = (x + y, y + z)$:

$T(x, y, z) = (0, 0)$

$x + y = 0 \Rightarrow x = -y$

$y + z = 0 \Rightarrow z = -y$

$\ker(T) = \{(-t, t, -t) : t \in \mathbb{R}\} = \text{span}\{(-1, 1, -1)\}$

Range (Image)

The range of $T: V \to W$ is:

$\text{range}(T) = \{T(\mathbf{v}) : \mathbf{v} \in V\} = \{\mathbf{w} \in W : \mathbf{w} = T(\mathbf{v}) \text{ for some } \mathbf{v}\}$

The set of all outputs of $T$.

Properties

• $\text{range}(T)$ is a subspace of $W$
• $\dim(\text{range}(T))$ = rank of $T$
• $T$ is surjective (onto) $\Leftrightarrow \text{range}(T) = W$

Example

For $T(x, y) = (x + y, x - y, 2x)$:

$T(x, y) = x(1, 1, 2) + y(1, -1, 0)$

$\text{range}(T) = \text{span}\{(1, 1, 2), (1, -1, 0)\}$

Dimension Theorem (Rank-Nullity)

For $T: V \to W$ where $V$ is finite-dimensional:

$\dim(V) = \dim(\ker(T)) + \dim(\text{range}(T)) = \text{nullity}(T) + \text{rank}(T)$

Example

$T: \mathbb{R}^3 \to \mathbb{R}^2$ with nullity = 1

$3 = 1 + \text{rank}(T)$

$\text{rank}(T) = 2$

So $\text{range}(T)$ has dimension 2 (could be all of $\mathbb{R}^2$).

For Matrix Transformations

If $T(\mathbf{x}) = A\mathbf{x}$:

Concept	Matrix Form
$\ker(T)$	Null space of $A$
$\text{range}(T)$	Column space of $A$
nullity	Number of free variables
rank	Number of pivot columns

Injectivity, Surjectivity, Bijectivity

Property	Condition	Kernel	Range
Injective	One-to-one	$\ker(T) = \{\mathbf{0}\}$	-
Surjective	Onto	-	$\text{range}(T) = W$
Bijective	Both	$\ker(T) = \{\mathbf{0}\}$	$\text{range}(T) = W$

Determining Properties

$T$ is Injective if:

• $\ker(T) = \{\mathbf{0}\}$
• $T(\mathbf{u}) = T(\mathbf{v}) \Rightarrow \mathbf{u} = \mathbf{v}$
• $\text{nullity}(T) = 0$

$T$ is Surjective if:

• $\text{range}(T) = W$
• $\text{rank}(T) = \dim(W)$
• Every $\mathbf{w} \in W$ has a preimage

$T$ is Bijective if:

• Both injective and surjective
• $T$ has an inverse $T^{-1}$

Example: Complete Analysis

$T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $T(x, y, z) = (x + y, y + z, x + 2y + z)$

Matrix form:

$A = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix}$

Row reduce:

$\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix}$

• rank = 2, nullity = 1
• $\ker(T) = \text{span}\{(1, -1, 1)\}$
• $\text{range}(T) = \text{span}\{(1, 0, 1), (1, 1, 2)\}$ (2-dimensional subspace of $\mathbb{R}^3$)
• Not injective ($\ker \neq \{\mathbf{0}\}$)
• Not surjective ($\text{range} \neq \mathbb{R}^3$)