Image and Kernel

Table of contents

1. Domain and Codomain
2. Image / Range
   1. Column Space
3. Kernel / Nullspace
   1. Homogeneous System
4. Rank-Nullity Theorem

---

Domain and Codomain

For $\Phi: V \rightarrow W$, $V$ is the domain and $W$ is the codomain of $\Phi$.

---

Image / Range

For $\Phi: V \rightarrow W$, the image or range of $\Phi$ is the set of all vectors in $W$ that are mapped from $V$.

$$ \mathrm{Im}(\Phi) = \{\Phi(\mathbf{x}) \in W \mid \mathbf{x} \in V\} $$

Also denoted as $\Phi(V)$.

Column Space

Let $A$ be the linear transformation matrix of $\Phi$.

Then the columns of $A$ span $\mathrm{Im}(\Phi)$:

\[\mathrm{Im}(\Phi) = \mathrm{span}[\mathbf{a}_1, \dots, \mathbf{a}_n]\]

Therefore $\mathrm{Im}(\Phi)$ is also called the column space of $A$.

• $\mathrm{rank}(A) = \mathrm{dim}(\mathrm{Im}(\Phi))$.

---

Kernel / Nullspace

For $\Phi: V \rightarrow W$, the kernel or nullspace of $\Phi$ is the set of all vectors in $V$ that are mapped to the zero vector in $W$.

$$ \mathrm{ker}(\Phi) = \{\mathbf{x} \in V \mid \Phi(\mathbf{x}) = \mathbf{0}\} $$

Also denoted as $\Phi^{-1}(\mathbf{0})$.

• Since $\Phi(\mathbf{0}) = \mathbf{0}$, kernel is never empty.
• If kernel is a singleton set of ${\mathbf{0}}$, then $\Phi$ is injective (one-to-one).
  • Also means the linear transformation matrix corresponding to $\Phi$ has linearly independent columns.

Homogeneous System

Let $A$ be the linear transformation matrix of $\Phi$.

Then $\mathrm{ker}(\Phi)$ is the solution set of the homogeneous system $A\mathbf{x} = \mathbf{0}$.

---

Rank-Nullity Theorem

Let $A$ be an $m \times \boldsymbol{n}$ linear transformation matrix of $\Phi: V \rightarrow W$.

So $V \in \mathbb{R}^n$ and $W \in \mathbb{R}^m$.

Then the rank-nullity theorem states that:

$$ \begin{gather*} \boldsymbol{n} = \rank(A) + \text{nullity}(A) \\[0.5em] \Updownarrow \\[0.5em] \dim(V) = \dim(\mathrm{Im}(\Phi)) + \dim(\ker(\Phi)) \end{gather*} $$

---