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# Theorem describing spaces associated to the block matrices of the extended reduced row echelon form of a matrix

Put content here**Theorem:** The extended RREF $[R \mid E]$ of $A$ simultaneously reveals all four fundamental subspaces:
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1. **Row space of $A$** = Row space of $R$ = span of the nonzero rows of $R$.
2. **Null space of $A$** = Null space of $R$, found by solving $Rx = 0$ using free variables.
3. **Column space of $A$** = span of the pivot columns of the original matrix $A$ (identified by pivot positions in $R$).
4. **Left null space of $A$** = span of the bottom $m - r$ rows of $E$.
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**Significance:** This is the most efficient single computation for understanding the complete structure of a linear transformation. All four subspaces and their dimensions are obtained from one row reduction.

# Parents

* Echelon matrices