Define the linear transformation L R2 rightarrow R2 by Le1

Define the linear transformation L: R^2 rightarrow R^2 by L(e_1) = 3e_1 + e_2 and L(e_2) = -e_1 + 3e_2. Find L^3(e_1) and L^3(e_2). (You don\'t need matrices.)

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. Then L³(e₁)= L(L²(e₁)) = L(8e₁+6e₂) = 8L(e₁)+ 6L(e₂)=8(3e₁+e₂)+6(-e₁+3e₂)= 24e₁ +8e₂ -6e₁ +18e₂ = 18e₁ +26e₂.

We have L²(e₂) = L(L(e₂))=L(-e₁+3e₂)=-L(e₁)+3L(e₂)= - (3e₁+e₂)+3(-e₁+3e₂)=-3e₁-e₂–3e₁+9e₂= -6e₁+8e₂. Then L³(e₂)= L(L²(e₂)) = L(-6e₁+8e₂) = -6L(e₁)+ 8L(e₂)=-6(3e₁+e₂)+8(-e₁+3e₂)= -18e₁-6e₂ -8e₁ +24e₂ = -26e₁ +18e₂.