It appears that you've uploaded an image of a math assignment. Let's go through each question with a focus on understanding and solving the problems, particularly for question 5 as requested. I'll guide you through finding the basis for the row, column, and null space of the matrix $A$.

The matrix given in question 5 is: $A = \begin{pmatrix} 1 & 2 & 0 & 2 & 5 \\ -2 & -5 & 1 & -1 & -8 \\ 0 & -3 & 3 & 4 & 1 \\ 3 & 6 & 0 & -7 & 2 \end{pmatrix}$

To find the basis for each space:

Row Space

1. Transform $A$ into row echelon form.
2. The non-zero rows in the row echelon form of $A$ will form the basis for the row space.

Column Space

1. Transform $A$ to reduced column echelon form or use the original matrix to identify pivot columns.
2. The pivot columns in the original matrix $A$ will form the basis for the column space.

Null Space

1. Set up the equation $A \mathbf{x} = 0$, where $\mathbf{x}$ is a vector of variables.
2. Solve this homogeneous system to find the solutions.
3. The solutions will span the null space, forming the basis for it.

Would you like me to proceed with the detailed steps for each calculation? Let me know if there's a specific part you'd like to explore first, or if you'd like a solution for another question on the assignment.

Here are five related questions to deepen your understanding:

1. How is the row echelon form useful for determining the rank of a matrix?
2. What is the relationship between the row space and the rank of a matrix?
3. Why do the pivot columns in the original matrix form the basis for the column space?
4. How do we interpret the dimension of the null space in terms of solutions to a linear system?
5. How does the concept of linear independence relate to finding a basis?

Tip: For matrices, remember that the rank (number of pivot columns) is a key indicator of the dimension for both row and column spaces, as well as the linear dependence among rows or columns.