21 - Row Rank

Row Rank (or Rank)

Row Rank or Rank of matrix $A$ is the number of non-zero rows in $\mathrm{RRE}(A)$
$\to \text{Notation: }\mathrm{Rank}(A)$

By Uniqueness of RRE Form then row rank is well-defined

Determining whether a system is consistent using row-rank / rank

Let $(A\mid \mathbf{b})$ be the matrix representing the linear system $A\mathbf{x}=\mathbf{b}$

$$\text{System is consistent}⟺\mathrm{Rank}(A\mid B)=\mathrm{Rank}(A)$$

Proof

Note this result was already demonstrated for systems in RRE form during the proof
Suppose that $\mathrm{RRE}(A)=PA$ where $P$ is a product of Elementary Matrices that reduce $A$
If $E$ is an elementary matrix then $\mathrm{RRE}(EA)=\mathrm{RRE}(A)$ by the uniqueness of $\mathrm{RRE}$ form
Hence

$$\mathrm{Rank}(EA)=\mathrm{Rank}(A)$$

Then

$$\begin{array}{rl}\mathrm{Rank}(A\mid \mathbf{b})=\mathrm{Rank}(A)& ⟺\mathrm{Rank}(PA\mid P\mathbf{b})=\mathrm{Rank}(PA)\\ & ⟺\text{the system }PA\mathbf{x}=P\mathbf{b}\text{ is consistent}\\ & ⟺\text{the system }A\mathbf{x}=\mathbf{b}\text{ is consistent}\end{array}$$

as the set of solutions to $A\mathbf{x}=\mathbf{b}$ is unaffected by EROs

Classifying the solutions of a Linear System

Let $A$ be a $m\times n$ matrix and $\mathbf{b}$ in ${\mathbb{R}}_{\mathrm{col}}^m$
1)

$$\text{no solutions}⟺(0,0,\cdots ,0\mid 1)\text{ is in }\mathrm{Row}(A\mid \mathbf{b})$$

$$\text{unique solution}⟺\mathrm{Rank}(A)=n$$

Also requires $m\ge n$

$$\text{infinitely many solutions}⟺\mathrm{Rank}(A)<n$$

Set of solutions is a $n\text{-}\mathrm{Rank}(A)$ parameter familyx

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Relation between $\mathrm{rowrank}$ and $\mathrm{colrank}$

From the definitions then $\mathrm{Col}(A)=\mathrm{Row}(A^T)$ so $\mathrm{colrank}(A)=\mathrm{rowrank}(A^T)$

Similarly
$\mathrm{Row}(A)=\mathrm{Col}(A^T)$ so $\mathrm{rowrank}(A)=\mathrm{colrank}(A^T)$