Exercise 5

Consider the vectors in ℝ³:

• u₁ = (1, 1, 0)
• u₂ = (0, 1, 1)
• u₃ = (1, 2, 1)

1. Give a basis of the vector subspace F = vect(u₁, u₂, u₃).
2. Show that F = { (x, y, z) ∈ ℝ³ ; x - y + z = 0 }.
3. Find a basis of the vector subspace G = { (x, y, z) ∈ ℝ³ ; x + y + z = 0 }.
4. Is F + G = ℝ³?
5. Give a basis of F ∩ G.
6. Complete the basis of G to form a basis of ℝ³.