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The problem from the image is asking about an orthogonal projection matrix onto a subspace \( V \) of \( \mathbb{R}^n \), and it specifically requires proving that the eigenvalues of this projection matrix \( P \) are either 0 or 1.

### Problem Breakdown:
1. **Given:**
   - \( V \) is a subspace of \( \mathbb{R}^n \).
   - \( \{ \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k \} \) is an orthonormal basis for \( V \).
   - \( P = \mathbf{V}\mathbf{V}^T \), where \( \mathbf{V} \) is the matrix formed by the orthonormal basis vectors as columns.

2. **To Prove:**
   - The eigenvalues of \( P \) are either 0 or 1.

3. **Hint Provided:**
   - You can write any \( x \in \mathbb{R}^n \) as \( \mathbf{x} = \mathbf{v} + \mathbf{v}^\perp \), where \( \mathbf{v} \in V \) and \( \mathbf{v}^\perp \in V^\perp \).

### Solution Outline:

1. **Orthogonal Projection Matrix Properties:**
   - The projection matrix \( P \) maps any vector \( \mathbf{x} \) onto the subspace \( V \).
   - If \( \mathbf{x} \in V \), then \( P\mathbf{x} = \mathbf{x} \), meaning \( P \mathbf{x} = 1 \cdot \mathbf{x} \).
   - If \( \mathbf{x} \in V^\perp \), the orthogonal complement of \( V \), then \( P \mathbf{x} = 0 \), meaning \( P \mathbf{x} = 0 \cdot \mathbf{x} \).

2. **Eigenvalue Analysis:**
   - For a vector \( \mathbf{x} \in V \), \( P \mathbf{x} = \mathbf{x} \). So, 1 is an eigenvalue of \( P \) with multiplicity equal to the dimension of \( V \).
   - For a vector \( \mathbf{x} \in V^\perp \), \( P \mathbf{x} = 0 \). So, 0 is an eigenvalue of \( P \) with multiplicity equal to the dimension of \( V^\perp \).

3. **Conclusion:**
   - Since any vector in \( \mathbb{R}^n \) can be decomposed into a part in \( V \) and a part in \( V^\perp \), and the action of \( P \) on these parts yields eigenvalues 1 and 0 respectively, the only possible eigenvalues for \( P \) are 0 and 1.

Would you like a more detailed step-by-step solution? Here are some related questions:

1. How do you find the orthonormal basis of a given subspace \( V \)?
2. What are the properties of orthogonal projection matrices?
3. How does the rank of the projection matrix \( P \) relate to the dimension of \( V \)?
4. Can you explain the spectral theorem and its relation to projection matrices?
5. How do you generalize this concept to other linear transformations?

**Tip:** Always remember that the sum of the eigenvalues (considering multiplicity) of a matrix equals its trace, and the product of the eigenvalues equals its determinant.

Suppose V is a subspace of R^n. Let {v1, v2, ..., vk} be an orthonormal basis for V. Let P = V(V^T). Prove that the eigenvalues of P are all either 0 or 1.

Learn how to prove that the eigenvalues of an orthogonal projection matrix P are either 0 or 1. This problem covers key linear algebra concepts, including orthogonal projections and eigenvalues, and is suitable for college-level students.

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