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Section 4.5 Rank-Nullity Theorem

We begin with a couple of definitions.
Definition 4.5.1.
Let \(V\) and \(W\) be vector spaces over a field \(F\text{.}\) Let \(T\colon V\to W\) be an \(F\)-linear map. The dimension of the kernel of \(T\) is called the nullity of \(T\text{.}\) So,
\begin{equation*} {\rm nullity}(T)=\dim_F\ker(T). \end{equation*}
Definition 4.5.2.
Let \(V\) and \(W\) be vector spaces over a field \(F\text{.}\) Let \(T\colon V\to W\) be an \(F\)-linear map. The dimension of the image of \(T\) is called the rank of \(T\text{.}\) So,
\begin{equation*} {\rm rank}(T)=\dim_F\Im(T). \end{equation*}

Let \(\{u_1,\ldots,u_r\}\) be a basis for \(\ker(T)\text{.}\) We extend this basis to a basis of \(V\text{,}\) say \(\{u_1,\ldots,u_r,v_1,\ldots,v_s\}\text{.}\) We claim that \(\{T(v_1),\ldots,T(v_s)\}\) is a basis for \(\Im(T)\text{.}\) We first check the independence. Suppose that

\begin{align*} 0\amp=\sum\alpha_iT(v_i)\\ \amp= T\big(\sum\alpha_iv_i\big) \end{align*}

Thus, \(\sum\alpha_iv_i\in\ker(T)\) and hence there exists \(\beta_j\in F\) such that

\begin{equation*} \sum\alpha_iv_i=\sum\beta_ju_j\quad\text{i.e.,}\quad\sum_i\alpha_iv_i-\sum_j\beta_ju_j=0. \end{equation*}

Since \(\{u_1,\ldots,u_r,v_1,\ldots,v_n\}\) is a basis we have \(\alpha_i=0\) and \(\beta_j=0\) for each \(i\) and \(j\text{.}\) Therefore, \(\{T(v_j)\}\) is linearly independent.

Given any \(w\in\Im(T)\) there exists \(v\in V\) such that \(w=T(v)\text{.}\) Let

\begin{equation*} v=\sum\gamma_iu_i+\sum\delta_jv_j. \end{equation*}

Using \(\{u_i\}\) is a basis of \(\ker(T)\) we have the following.

\begin{align*} w\amp=T(v)\\ \amp=\sum\gamma_iT(u_i)+\sum\delta_jT(v_j)\\ \amp=\sum\delta_jT(v_j) \end{align*}

Hence \(\{T(v_j)\}\) spans \(\Im(T)\text{.}\) Since \(\{T(v_j)\}\) is linearly independent and spans the \(\Im(T)\) it is a basis of \(\Im(T)\text{.}\) We thus have

\begin{equation*} \dim_F\Im(T)=s\quad\text{and}\quad\dim_F\ker(T)=r. \end{equation*}

Therefore,

\begin{equation*} \dim_FV={\rm nullity}(T)+{\rm rank}(T). \end{equation*}