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The problem asks to find \( u = (u_1, u_2, u_3) \) and \( v = (v_1, v_2, v_3) \) such that the matrix 

\[
A = \begin{bmatrix} 
1 & u_1 & 0 & v_1 & 1 \\
2 & u_2 & 2 & v_2 & 1 \\
1 & u_3 & 1 & v_3 & 0 
\end{bmatrix}
\]

has a null space \( N(A) = \text{span} \left\{ (0, 2, 1, 1, 0), (-1, 2, 0, 2, 1) \right\} \).

### Approach:

We need to find the vectors \( u = (u_1, u_2, u_3) \) and \( v = (v_1, v_2, v_3) \) such that the given null space holds.

#### Step 1: Null space condition
If \( N(A) \) is the null space, this means that for each vector in the null space, say \( x = (x_1, x_2, x_3, x_4, x_5) \), it must satisfy \( A x = 0 \). Hence, we will set up the system for each of the null vectors:
1. \( (0, 2, 1, 1, 0) \)
2. \( (-1, 2, 0, 2, 1) \)

#### Step 2: Apply each null vector
For each null vector, compute \( A \cdot x = 0 \) to form equations that involve \( u_1, u_2, u_3 \) and \( v_1, v_2, v_3 \).

Let me calculate and solve the system.The solution for \( u = (u_1, u_2, u_3) \) and \( v = (v_1, v_2, v_3) \) such that the matrix \( A \) has the given null space is:

\[
u = (u_1, u_2, u_3) = \left( 0, -\frac{5}{2}, -\frac{3}{2} \right)
\]
\[
v = (v_1, v_2, v_3) = (0, 3, 2)
\]

This ensures that the null space \( N(A) = \text{span} \{ (0, 2, 1, 1, 0), (-1, 2, 0, 2, 1) \} \).

Would you like further clarification or details?

Here are 5 related questions to further explore this concept:
1. How do we interpret the null space of a matrix geometrically?
2. What is the rank-nullity theorem, and how does it apply to this matrix?
3. How do you determine the dimension of a matrix's null space?
4. Can a matrix have multiple null spaces, or is it unique?
5. What does it mean if a matrix has a trivial null space?

**Tip:** In general, the null space of a matrix consists of all the vectors that map to the zero vector when multiplied by the matrix.

Find u = (u1, u2, u3) and v = (v1, v2, v3) such that the matrix A has the null space N(A) = span{(0, 2, 1, 1, 0), (-1, 2, 0, 2, 1)}.

Solve for u = (u1, u2, u3) and v = (v1, v2, v3) such that the matrix A has the null space span{(0, 2, 1, 1, 0), (-1, 2, 0, 2, 1)}. This problem involves key concepts from linear algebra including matrices, null spaces, and the rank-nullity theorem.

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