Dashboard

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

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

Created over 8 years ago, updated 10 days ago

Theorem: The extended RREF $[R \mid E]$ of $A$ simultaneously reveals all four fundamental subspaces:

  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 $E$.

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.