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The row space of a matrix A is the vector subspace spanned by its linearly independent row vectors (that is, those that are not zero when in reduced row echelon form). Unlike the column space, the basis of the row space is not affected by elementary row operations. For example, the row space of the following matrix, which can be row-reduced to the second matrix,

$ \begin{bmatrix} -3 & 2 & 1 \\ 4 & -2 & 2 \\ 6 & -4 & -2 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & 5 \\ 0 & 0 & 0 \end{bmatrix} $

will be spanned by both

$ \{ \begin{bmatrix} -3 & 2 & 1 \end{bmatrix}, \begin{bmatrix} 4 & -2 & 2 \end{bmatrix} \} $

and

$ \{ \begin{bmatrix} 1 & 0 & 3 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 5 \end{bmatrix} \} $

The row space can also be found by finding the column space of the transpose of A. By the rank theorem, the dimension of the row space of A will be equal to the dimension of the column space of A, which will be the rank of the matrix.

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