Find the basis and dimension for the row space of the matrix $$A=\left[\begin{array}{ccc}2& 4& -2\\ 1& 2& -1\\ 3& 6& -3\end{array}\right]$$

Step 1 ：Step 1: We begin by row reducing the given matrix. We can subtract half of the first row from the second row and subtract the first row from the third row to get: $$\left[\begin{array}{ccc}2& 4& -2\\ 0& 0& 0\\ 1& 2& -1\end{array}\right]$$

Step 2 ：Step 2: To simplify further, we can divide the first row by 2 and subtract it from the third row to get: $$\left[\begin{array}{ccc}1& 2& -1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$

Step 3 ：Step 3: We can see that the row reduced matrix has only one non-zero row. So, the basis for the row space of the matrix is the set of vectors formed by the non-zero rows. Hence, the basis of the row space of matrix A is: $$\left\{\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]\right\}$$

Step 4 ：Step 4: The dimension of the row space is the number of vectors in its basis. So, the dimension of the row space of matrix A is 1.