3.5: Row Space

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 $\mathscr{Ra}⁢(A^{T})$.