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Section 6.2 The Kernel and range of \(T\)
Definitions of kernel and range.
Let \(T\) be a linear transformation from \(V\) to \(W\text{.}\)
Kernel of \(T:\)
\begin{equation*}
\operatorname{ker}(T)=\{v\in V|T(v)=\mathbf{0}_W\}.
\end{equation*}
Range of \(T:\)
\begin{align*}
\operatorname{ran}(T)=\amp\{T(v)|v\in V\} \\
=\amp\{w\in W|\exists v\in V\text{ such that } w=T(v)\}
\end{align*}
Theorem 6.2.1.
\(\operatorname{ker}(T)\) is a subspace of \(V\text{.}\)
\(\operatorname{ran}(T)\) is a subspace of \(W\text{.}\)
For the rest of this section, we assume that \(T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}.\) Recall that \(\forall v\in V\text{,}\)
\begin{equation*}
T(v)=Av\text{,}
\end{equation*}
where \(A=\Big[T(\mathbf{e}_1)\, T(\mathbf{e}_2)\,\ldots\, T(\mathbf{e}_n)\Big]\text{.}\)
Theorem 6.2.2.
Let A be the standard matrix of the linear transformation \(T\text{.}\)
\(\displaystyle \operatorname{ker}(T)=\operatorname{Nullspace}(A).\)
\(\displaystyle \operatorname{ran}(T)=\operatorname{col}(A).\)
Proof.
-
\(\forall v\in \operatorname{ker}(T), T(v)=\mathbf{0}.\) It follows from \(T(v)=Av\) that \(Av=\mathbf{0}.\) Thus \(v\in \operatorname{Nullspace}(A).\) Therefore \(\operatorname{ker}(T)\subseteq \operatorname{Nullspace}(A).\)
\(\forall v\in \operatorname{Nullspace}(A), \) \(Av=\mathbf{0}\) it follows from \(T(v)=Av\) that \(T(v)=\mathbf{0}.\) Thus \(v\in \operatorname{Ker}(T).\) Therefore \(\operatorname{Nullspace}(A)\subseteq \operatorname{ker}(T).\)
Therefore, \(\operatorname{ker}(T)=\operatorname{Nullspace}(A).\)
It follows from
Theorem 6.2.2 that the kernel and range of
\(T\) is related closely with the standard matrix
\(A\text{.}\)
