Exercise 3

Let (e₁, e₂, e₃) be the canonical basis of ℝ³ and u an endomorphism of ℝ³ defined by:

u(e₁) = 2e₁ + e₂ + 3e₃  
u(e₂) = e₂ − 3e₃  
u(e₃) = −2e₂ + 2e₃

1. Let x = (x₁, x₂, x₃) ∈ ℝ³. Determine u(x).
2. Let:

   E = {x ∈ ℝ³ | u(x) = 2x}  
   F = {x ∈ ℝ³ | u(x) = −x}
       

   Show that E and F are two vector subspaces of ℝ³.
3. Find a basis of E and a basis of F.
4. Do we have E ⊕ F = ℝ³?

Exercise 4

Let f : ℝ³ → ℝ³ be a linear map defined by:

f(x, y, z) = (2x + y + z, x + 2y + z, 4x − y + z)

1. Give a basis of ker f and a basis of Im f. Is the map f invertible?
2. Verify the relation: dim(ℝ³) = dim(ker f) + dim(Im f).
3. Determine f²(x, y, z).