Dashboard

Now you are in the subtree of Container for Linear Algebra project. 

Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrix

Created over 8 years ago, updated 10 days ago

Theorem: Let $[A \mid I] \to [R \mid E]$ be the extended RREF of an $m \times n$ matrix $A$ of rank $r$. Then the block matrix $E$ has the following structure:

  • The first $r$ rows of $E$ give the coefficients that express the pivot rows of $R$ as linear combinations of the rows of $A$.
  • The last $m - r$ rows of $E$ form a basis for the left null space of $A$ (i.e., the null space of $A^T$).

Explanation: Since $EA = R$ and the last $m - r$ rows of $R$ are zero, the corresponding rows of $E$ satisfy $E_{\text{bottom}} A = 0$, meaning they are in the left null space.