Image and Kernel

For $\Phi:V\rightarrow W$ , we define the kernel/null space

\begin{align} \text{ker}(\Phi) \coloneqq \Phi^{-1}(\boldsymbol{0}_{W}) = \{v\in V:\Phi(v)=\boldsymbol{0}_{W}\} \tag{1} \end{align}

and the image/range

\begin{align} \text{Im}(\Phi) \coloneqq \Phi(V) = \{\boldsymbol{w\in W|\exists \boldsymbol{v}\in V:\Phi(\boldsymbol{v}=\boldsymbol{w})}\}.\tag{2} \end{align}

We also call $V$ and $W$ as domain and codomain of $\Phi$ .

Consider a linear mapping $\Phi : V \rightarrow W$ , where $V$ , $W$ are vector spaces.

It always holds that $\Phi(\mathbf{0}_{V})=\mathbf{0}_{W}$ and, therefore, $\mathbf{0}_{V}\in \text{ker}(\Phi)$ . In particular, the null space is never empty.
$\text{Im}(\Phi) \subseteq W$ is a subspace of $W$ , and $\text{ker}(\Phi) \subseteq V$ is a subspace of $V$ .
$\Phi$ is injective (one-to-one) if and only if $\text{ker}(\Phi)=\{0\}$ . (Null Space and Column Space). Let us consider $A \in \mathbb{R}^{m×n}$ and a linear mapping $\Phi:\mathbb{R}^n\rightarrow\mathbb{R}^m$ , $x\mapsto\boldsymbol{A}x$ .
For $\boldsymbol{A} = [a_{1}, \dots,a_{n}]$ , where $a_i$ are columns of $\boldsymbol{A}$ , we obtain

\begin{align} \text{Im}(\Phi) = \{\boldsymbol{A}x:x\in\mathbb{R}^n\}=\left\{\sum_{i=1}^nx_{i}a_{i}:x_{1},\dots,x_{n}\in \mathbb{R}\right\} \tag{3} \\ =\text{span}[a_{1},\dots, a_{n}] \subseteq \mathbb{R}^m \tag{4} \end{align}

$\text{rk}(\boldsymbol{A})$ = $\text{dim(Im(}\Phi))$ .
The kernel/null space $\text{ker}(\Phi)$ is the general solution to the homogeneous system of linear equations Ax = 0 and captures all possible linear combinations of the elements in Rn that produce 0 ∈ Rm. The kernel is a subspace of Rn , where n is the “width” of the matrix.

For vector spaces $V$ , $W$ and a linear mapping $\Phi:V\rightarrow W$ it holds that

\begin{align} \text{dim}(\text{ker}(\Phi)) + \text{dim}(\text{Im}(\Phi)) = \text{dim}(V). \tag{5} \end{align}