Define the linear transformation L R2 R2 by Le1 3e1 e2 an

Define the linear transformation L: R^2 --> R^2 by L(e1) = 3e1 + e2 and L(e2) = -e1 + 3e2. Find L^2(e1) and L^2(e2). (YOU DONT NEED MATRICIES)

Solution

We have L²(e₁) = L(L(e₁ )) = L(3e₁ + e₂) = 3L( e₁)+L( e₂) = 3(3e₁ + e₂)+( -e₁ + 3e₂) = 9e₁ +3e₂ –e₁ +3e₂ =          8e₁ +6e₂. Further, L²(e₂) = L(L(e₂ )) = L(-e₁ + 3e₂) = -L(e₁)+3L(e₂ ) = -(3e₁ + e₂) +3(-e₁ + 3e₂) = -3e₁ - e₂ - 3e₁ + 9e₂ = -6e₁+8e₂.

Note: L, being a linear transformation, preserves vector addition and scalar multiplication.