```markdown
### Eliminating Equality Constraints

The slide discusses the representation of solutions to the equation \( \{ x \mid Ax = b \} \) as:

\[ \{ x \mid Ax = b \} = \{ Fz + \hat{x} \mid z \in \mathbb{R}^{n-p} \} \]

- **Key Points:**
  - \( \hat{x} \) is any particular solution.
  - The range of \( F \in \mathbb{R}^{n \times (n-p)} \) is the nullspace of \( A \) (with \( \text{Rank} \, F = n-p \) and \( AF = 0 \)).

- **Reduced or Eliminated Problem:**
  - Minimize \( f(Fz + \hat{x}) \).

- **Additional Notes:**
  - This is an unconstrained problem with variable \( z \in \mathbb{R}^{n-p} \).
  - From solution \( z^\star \), obtain \( x^\star \) and \( \nu^\star \) as:
    \[
    x^\star = Fz^\star + \hat{x}, \quad \nu^\star = -(A A^T)^{-1} A \nabla f(x^\star)
    \]

*Source: A. d'Aspremont. Convex Optimization M2.*
```
