Row and column spaces: Difference between revisions

Revision as of 18:40, 1 December 2018

Revision as of 02:58, 1 August 2019

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Note that the independent columns of the reduced row echelon form are precisely the columns with [[Pivot element|pivots]]. This makes it possible to determine which columns are linearly independent by reducing only to [[row echelon form|echelon form]].

The above algorithm can be used in general to find the dependence relations between any set of vectors, and to pick out a basis from any spanning set. A different algorithm for finding a basis from a spanning set is given in the [[row space]] article; Also finding a basis for the column space of ''A'' is equivalent to finding a basis for the row space of the [[transpose]] matrix ''A''^T.

To find the basis in a practical setting (e.g., for large matrices), the [[singular-value decomposition]] is typically used.