The matrix provided, along with its reduced row echelon form (RREF), allows us to identify the bases for the column space (Col A) and the null space (Nul A) of the matrix $A$.

Given the matrix:

$A = \begin{bmatrix} -1 & 0 & -4 & -2 \\ -7 & 3 & -13 & -17 \\ -2 & 1 & -3 & -5 \end{bmatrix} \sim \begin{bmatrix} 1 & 0 & 4 & 2 \\ 0 & 1 & 5 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

Analysis:

1. Column Space (Col A):

   • The columns corresponding to the pivot positions in the RREF are the basis vectors for Col A in the original matrix.
   • From the RREF, the pivots are in the first and second columns.
   • Therefore, the basis for Col A will consist of the first and second columns of the original matrix $A$: $\left\{ \begin{bmatrix} -1 \\ -7 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} \right\}$

2. Null Space (Nul A):

   • To find a basis for Nul A, we solve for the solutions to $A \mathbf{x} = 0$.
   • By setting free variables based on the non-pivot columns (columns 3 and 4 in this case), we can find vectors that span the null space.

It looks like you've already determined a correct basis for Nul A as: $\left\{ \begin{bmatrix} -4 \\ -5 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \\ 0 \\ 1 \end{bmatrix} \right\}$

Summary:

• Basis for Col A: $\left\{ \begin{bmatrix} -1 \\ -7 \\ -2 \end{bmatrix}, \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix} \right\}$
• Basis for Nul A: $\left\{ \begin{bmatrix} -4 \\ -5 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -2 \\ 1 \\ 0 \\ 1 \end{bmatrix} \right\}$

Would you like further explanation on any of these steps?

Here are five related questions you may find helpful:

1. How do you determine the rank of a matrix from its RREF?
2. Why do the pivot columns in the original matrix form a basis for the column space?
3. What is the relationship between the rank and the number of vectors in the null space basis?
4. How does the dimension of the column space relate to the row space?
5. How can we verify that a set of vectors forms a basis for a space?

Tip: When determining the null space, focus on identifying free variables from the RREF, as they directly determine the vectors that span the null space.