\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
In \(\mathbb{R}^3\), set \(F = \mathrm{Vect}\bigl((1, 0, 0)\bigr)\), \(G = \mathrm{Vect}\bigl((0, 1, 0)\bigr)\), \(H = \mathrm{Vect}\bigl((1, 1, 0)\bigr)\). Show that
  1. \(F \cap G = \{0\}\), \(F \cap H = \{0\}\) and \(G \cap H = \{0\}\) (pairwise trivial intersections);
  2. \(F, G, H\) are not in direct sum.

Capture an image of your work. AI teacher feedback takes approximately 10 seconds.