Row and column spaces: Difference between revisions

Revision as of 21:14, 24 September 2019

Revision as of 10:49, 3 November 2019

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\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}.

</math>

Once the matrix is in echelon form, the nonzero rows are a basis for the row space. In this case, the basis is {&nbsp;(1,&#8239;3,&#8239;2),&nbsp;(02,&#8239;17,&#8239;04)&nbsp;}. Another possible basis {&nbsp;(1,&#8239;0,&#8239;2),&nbsp;(0,&#8239;1,&#8239;0)&nbsp;} comes from a further reduction.<ref name="example">The example is valid over the [[real number]]s, the [[rational number]]s, and other [[number field]]s. It is not necessarily correct over fields and rings with non-zero [[characteristic (algebra)|characteristic]].</ref>

This algorithm can be used in general to find a basis for the span of a set of vectors. If the matrix is further simplified to [[reduced row echelon form]], then the resulting basis is uniquely determined by the row space.