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The question asks about a 3x3 matrix \( A \) with eigenvalues \( 0, 3, 5 \) and corresponding eigenvectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \). It seeks to determine when the linear system \( A\mathbf{x} = 3\mathbf{v} + 5\mathbf{w} \) has infinitely many solutions.

### Key Concepts:
1. **Eigenvalue-Eigenvector Property**:
   If \( A\mathbf{v} = \lambda\mathbf{v} \), then \( \mathbf{v} \) is an eigenvector of \( A \) corresponding to the eigenvalue \( \lambda \).

2. **Linearity of the System**:
   For the system \( A\mathbf{x} = 3\mathbf{v} + 5\mathbf{w} \), the right-hand side must belong to the column space of \( A \) for the system to have a solution.

3. **Rank and Infinitely Many Solutions**:
   A system of equations has infinitely many solutions if it is consistent (i.e., the right-hand side belongs to the column space of \( A \)) and the rank of \( A \) is less than the dimension of the vector space (indicating free variables).

### Solution Steps:
1. **Eigenvalue \( 0 \) and Its Effect**:
   The eigenvalue \( 0 \) implies that \( A \) is not invertible and has a null space (kernel) of dimension at least 1. This creates the potential for infinitely many solutions if the system is consistent.

2. **Condition for Consistency**:
   For the system \( A\mathbf{x} = 3\mathbf{v} + 5\mathbf{w} \) to be consistent, the vector \( 3\mathbf{v} + 5\mathbf{w} \) must lie in the column space of \( A \). Since \( \mathbf{v} \) and \( \mathbf{w} \) are eigenvectors with non-zero eigenvalues, \( 3\mathbf{v} + 5\mathbf{w} \) lies in the column space of \( A \).

3. **Conclusion**:
   Since the matrix \( A \) has eigenvalue \( 0 \), the rank of \( A \) is 2 (not full rank), and the system is consistent. Therefore, the system \( A\mathbf{x} = 3\mathbf{v} + 5\mathbf{w} \) has infinitely many solutions.

Let me know if you’d like further clarification or a detailed breakdown! 

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### Related Questions:
1. What is the role of the eigenvalue \( 0 \) in determining the solution space of \( A\mathbf{x} = \mathbf{b} \)?
2. How can we confirm whether \( 3\mathbf{v} + 5\mathbf{w} \) is in the column space of \( A \)?
3. What are the geometric implications of the null space in a 3x3 matrix with eigenvalue \( 0 \)?
4. How does the rank-nullity theorem apply to this matrix \( A \)?
5. Can we generalize this result to an \( n \times n \) matrix with eigenvalue \( 0 \)?

### Tip:
Always check the rank of the matrix and the consistency of the system before concluding whether solutions exist and if they are unique or infinite.

If A is a 3 x 3 matrix with eigenvalues 0, 3, 5 and corresponding eigenvectors u, v, w, then the linear system Ax = 3v + 5w has infinitely many solutions.

Explore the solution of a linear system involving a 3x3 matrix with eigenvalues 0, 3, and 5. Understand key linear algebra concepts like eigenvalues, eigenvectors, and the rank-nullity theorem. Suitable for undergraduate students studying linear algebra.

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