\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Compute the dimension of each of the following subspaces.
  1. \(F_1 = \{(x, y, z) \in \mathbb{R}^3 : x + y + z = 0\}\).
  2. \(F_2 = \{P \in \mathbb{R}_3[X] : P(1) = 0\}\).
  3. \(F_3 = \{M \in \mathcal{M}_2(\mathbb{R}) : \mathrm{tr}(M) = 0\}\) (traceless matrices).

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