Kernel and Range

Definition

Let $T_A:R^n\to R^m$ : Matrix transformation

If $\ker (T_A)$ is a set of all vectors in $R^n$ that $T_A$ maps to $\{0\}$

Then

• We call $\ker (T_A)$ the kernel of $T_A$
• $\forall \mathbf{x}\in (\ker (T_A)\subseteq R^n):\mathbf{x}\stackrel{T_A}{\to }\{0\}$

If $R(T_A)$ is the set of all vectors in $R^m$ that are images under $\mathbf{x}\stackrel{T_A}{\to }\{0\}$ of at least one vector in $R^n$

Then we call $R^n$ the range of $T_A$

Note

In brief:

$$\begin{array}{rl}\ker (T_A)& =\text{null space of }A\\ R(T_A)& =\text{column space of }A\end{array}$$