Given the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix}$, find the basis and dimension for the row space of the matrix.

Step 1 ：Step 1: Perform row operations to bring the matrix to row-echelon form. This does not change the row space. We subtract twice the first row from the second, and thrice the first row from the third to get: $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

Step 2 ：Step 2: The row space of this matrix is the span of the non-zero rows. So, the basis for the row space of the original matrix is the set $\{\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\}$

Step 3 ：Step 3: The dimension of the row space is the number of vectors in the basis. So, the dimension of the row space of the matrix is 1.