5.2.5: Row Space

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The Row Space

As the columns of \(A^{T}\) are simply the rows of \(A\) we call \(Ra(A^{T})\) the row space of \(A^{T}\). More precisely

Definition: Row Space

The row space of the m-by-n matrix A is simply the span of its rows, i.e.,

\[Ra(A^{T}) \equiv \{A^{T} \textbf{y} | \textbf{y} \in \mathbb{R}^{m}\} \nonumber\]

This is a subspace of \(\mathbb{R}^n\)

Let us examine the matrix:

\[A = \begin{pmatrix} {0}&{1}&{0}&{0}\\ {-1}&{0}&{1}&{0}\\ {0}&{0}&{0}&{1} \end{pmatrix} \nonumber\]

The row space of this matrix is:

\[\mathscr{Ra}(A^{T}) = \left \{ y_{1} \begin{pmatrix} {0}\\{1}\\{0}\\{0} \end{pmatrix}+y_{2} \begin{pmatrix} {-1}\\{0}\\{1}\\{0} \end{pmatrix}+y_{3} \begin{pmatrix} {0}\\{0}\\{0}\\{1} \end{pmatrix} | y \in \mathbb{R}^{3} \right \} \nonumber\]

As these three rows are linearly independent we may go no further. We "recognize" then \(\mathcal{Ra}(A^{T})\) as a three dimensional subspace of \(\mathbb{R}^{4}\)

Method for Finding the Basis of the Row Space

Regarding a basis for \(\mathscr{Ra}(A^T)\) we recall that the rows of \(A_{red}\), the row reduced form of the matrix \(A\), are merely linear \(A\) combinations of the rows of \(A\) and hence

\[\mathscr{Ra}(A^T) = \mathscr{Ra}(A_{red}) \nonumber\]

This leads immediately to:

Definition: A Basis for the Row Space

Suppose \(A\) is m-by-n. The pivot rows of \(A_{red}\) constitute a basis for \(\mathscr{Ra}⁢(A^{T})\).

With respect to our example,

\[\left \{ \begin{pmatrix} {0}\\{1}\\{0}\\{0} \end{pmatrix}, \begin{pmatrix} {-1}\\{0}\\{1}\\{0} \end{pmatrix}, \begin{pmatrix} {0}\\{0}\\{0}\\{1} \end{pmatrix} \right \} \nonumber\]

comprises a basis for \(\mathscr{Ra}⁢(A^{T})\).