[Engineering Mathematics] 9. Rank Theorem

Definition. The column rank of an \(m \times n\) matrix \(\boldsymbol{A}\) is \(\text{rank} \boldsymbol{A}=\text{dim}(\text{span} S)\), where \(S=\{\boldsymbol{a}_1, ..., \boldsymbol{a}_n\}\). \(\text{span} S\) is called the column space.

 

Rank Theorem. The column rank is the same as the row rank.

 

Proof Outline

Elementary row operations do not change the dimension of the column space and the row space. Therefore, column rank and row rank are both the same as the number of leading-\(1\)s in the RREF form of the matrix.

\(\square\)