Row and column spaces: Difference between revisions

Revision as of 08:44, 9 August 2019

Revision as of 02:59, 10 August 2019

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===Relation to the left null space===

The [[left null space]] of ''A'' is the set of all vectors '''x''' such that '''x'''^T''A^T'' = '''0'''^T. It is the same as the [[kernel (matrix)|null space]] of the [[transpose]] of ''A''. The product of the matrix ''A''^T and the vector '''x''' can be written in terms of the [[dot product]] of vectors:

:<math>A^\mathsf{T}\mathbf{x} = \begin{bmatrix} \mathbf{v}_1 \cdot \mathbf{x} \\ \mathbf{v}_2 \cdot \mathbf{x} \\ \vdots \\ \mathbf{v}_n \cdot \mathbf{x} \end{bmatrix},</math>

because [[row vector]]s of ''A''^T are transposes of column vectors '''v'''_''k'' of ''A''. Thus ''A''^T'''x'''&nbsp;=&nbsp;'''0''' if and only if '''x''' is [[orthogonal]] (perpendicular) to each of the column vectors of ''A''.