Kernel and Image in linear algebra

Kernel

For a linear transformation $T$ $\mathrm{Ker}(T):=\{\mathbf{x}\in {\mathbb{R}}^n:T(x)=0\}\subseteq {\mathbb{R}}^n$

$$\mathrm{Ker}(T)=\mathrm{N}(A)$$

NOTE

The kernel of a matrix is the null space of it

Properties

• $\ker (A)$ is a subspace of ${\mathbb{R}}^n$.
• The nullity is $\dim (\ker (A))$.
• Rank–Nullity Theorem:

$$\dim (\ker (A))+\text{rank}(A)=n$$ Link to original

Image

For a linear transformation $T$ $\mathrm{Im}(T):=\{T(\mathbf{x}):\mathbf{x}\in {\mathbb{R}}^n\}\subseteq {\mathbb{R}}^m$

In matrix

$$\mathrm{Im}(T)=\mathrm{C}(A)$$