T:V → V, where V is a vector space over R

T(x) = 0 ; x ∈ V

T is linear

T:R → R²

T(x) = (x, 3x); x ∈ R

T is linear

T:R² → R²

T(x, y) = (x, 0); x, y ∈ R

T is linear

T:R² → R²

T(x) = (1, 1); x ∈ R

T is not linear

[∵ T(0) ≠ (0, 0)]

T:R → R²

T(x) = (x, 3x+1); x ∈ R

T is not linear

[∵ T(0) ≠ (0, 0)]

T:R → R²

T(x) = (x, x²); x ∈ R

T is not linear

  [T(1+2) = T(3) = (3, 9)
and  T(1)+T(2) = (1, 1)+(2, 4) =(3, 5)
⇒ T(1+2) ≠ T(1)+T(2)