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

Put content here**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:
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- 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 $A^T$).
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**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.

# Parents

* Echelon matrices