To reverse engineer a matrix $A$ given the span of its null space, we start by recognizing that the null space vectors form a basis for the null space of $A$. In other words, each of these vectors is a solution to the homogeneous system $A \mathbf{x} = 0$.

The two null space vectors are:

• $\mathbf{v}_1 = (-2, -1, 1, 0, 0)$
• $\mathbf{v}_2 = (-3, 2, 0, -2, 1)$

These vectors tell us that: $A \mathbf{v}_1 = 0 \quad \text{and} \quad A \mathbf{v}_2 = 0$

To reverse engineer the matrix $A$, we can construct $A$ by creating a matrix whose rows are orthogonal to both $\mathbf{v}_1$ and $\mathbf{v}_2$. Since the null space is spanned by these two vectors, $A$ must have at least three rows (because the dimension of the null space is 2 and $A$ is underdetermined with respect to the number of constraints).

Step-by-Step Construction:

1. Write the null space vectors as columns of a matrix:

   \begin{pmatrix} -2 & -3 \\ -1 & 2 \\ 1 & 0 \\ 0 & -2 \\ 0 & 1 \end{pmatrix}$$ This matrix $$N$$ represents the null space of the matrix $$A$$.

2. Construct $A$ by finding a matrix whose row space is orthogonal to the column space of $N$:

   To find $A$, we need to compute a matrix $A$ such that $A \mathbf{v}_1 = 0$ and $A \mathbf{v}_2 = 0$, which means each row of $A$ must be orthogonal to both $\mathbf{v}_1$ and $\mathbf{v}_2$.

   We can solve for this by finding vectors orthogonal to both null space vectors.

3. Compute the matrix $A$: Let's consider the orthogonal complement of the null space. For simplicity, we assume that $A$ is a $3 \times 5$ matrix (since the null space has dimension 2, the row space must have dimension 3).

   One possible matrix $A$ could be constructed using the row reduction process or by finding the orthogonal complement directly. Here's a potential form of the matrix:

   \begin{pmatrix} 1 & 0 & 2 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 \end{pmatrix}$$ This matrix has the required property that: $$A \mathbf{v}_1 = 0 \quad \text{and} \quad A \mathbf{v}_2 = 0$$

Final Answer:

The matrix $A$ whose null space is spanned by the vectors $(-2, -1, 1, 0, 0)$ and $(-3, 2, 0, -2, 1)$ is:

\begin{pmatrix} 1 & 0 & 2 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 \end{pmatrix}$$ Let me know if you need more details on this process. --- Would you like further explanation on this? Here are 5 related questions to explore: 1. What is the significance of the null space in linear algebra? 2. How do you find the orthogonal complement of a vector space? 3. What is the relationship between the rank of a matrix and the null space? 4. Can the null space determine the uniqueness of the matrix $$A$$? 5. How can the null space help in solving systems of linear equations? **Tip:** The dimension of the null space is called the "nullity" of the matrix, and it provides important information about the number of solutions to the system $$A\mathbf{x} = 0$$.