Exercise 1

Let f : ℝ³ → ℝ² be defined by:

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

1. Show that f is linear.
2. Give a basis of the kernel of f, and deduce rank(f).

Exercise 2

Let (e₁, e₂) and (e′₁, e′₂, e′₃) be the canonical bases of the vector spaces ℝ² and ℝ³ respectively.

1. Determine the linear map f : ℝ² → ℝ³ such that:

   f(e₁) = 2e′₁ + e′₃, f(e₂) = −5e′₁ + e′₂

2. Determine ker f, Im f, and specify their dimensions.
3. Is the map f injective? Surjective? Justify.