Given a matrix A = $\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$, find the basis and dimension for the column space of the matrix.

Step 1 ：First, we perform the Gaussian elimination to row reduce the matrix to its row echelon form. The row echelon form of the matrix A is $\left[\begin{array}{ccc}1& 2& 3\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$.

Step 2 ：The column space of the matrix A is spanned by the columns of A that correspond to the leading 1's in the row echelon form. In this case, only the first column has a leading 1, so the column space is spanned by the first column of A.

Step 3 ：Therefore, the basis for the column space of the matrix A is $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$.

Step 4 ：The dimension of the column space of a matrix is the number of vectors in its basis. Thus, the dimension for the column space of the matrix A is 1.