Determine the column space for the matrix \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]

Step 1 ：The column space of a matrix is the set of all possible linear combinations of its column vectors. The column vectors of matrix A are \[ \vec{a} = \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix}, \vec{b} = \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix}, \vec{c} = \begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix} \]

Step 2 ：The column space is the set of all vectors that can be written in the form \[ c_1 \vec{a} + c_2 \vec{b} + c_3 \vec{c} \] where \(c_1, c_2, c_3\) are scalars.

Step 3 ：Since the column vectors are linearly dependent (they are multiples of each other), the column space of this matrix is the line spanned by any one of the column vectors. For instance, \[\vec{a} = \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix}\]