Theorem: Let **V** be a **vector space**. Let **R**, **S**, and **T** ∈ **L**(**V**) be **linear transformations**. Then:

(a) R(S + T) = RS + RT and (S + T)R = SR + TR

(b) R(ST) = (RS)T

(c) RI = IR = R

(d) c(ST) = (cS)T = S(cT) ∀ c ∈ **R**

-----------------

Let R,S,T,W ∈ L(V) be linear transformations, let u = {u_1, u_2} and v = {v_1, v_2}, and let c ∈ **R**. Let I be the multiplicative identity transformation such that IL = LI = L ∀ L ∈ L(V). Then:

R(S(cu) + T(v)) = IR(S(cu) + T(v))

= RI(S(cu_1, cu_2) + T(v_1, v_2)) = RcIS(u_1, u_2) + RIT(v_1, v_2)

= cRS(u_1, u_2)I + RT(v_1, v_2)I = cRS(u) + RT(v).

Similarly,

(S + cR)(T(u)W(v)) = (S + cR)(T(u_1, u_2)W(v_1, v_2))

= S(T(u_1, u_2)W(v_1, v_2)) + cR(T(u_1, u_2)W(v_1, v_2))

= (ST(u_1, u_2))(W(v_1, v_2)) + (RcT(u_1, u_2))(W(v_1, v_2))

= (ST(u_1, u_2)W(v_1, v_2)) + (RT(cu_1, cu_2))W(v_1, v_2)

= ST(u_1, u_2)W(v_1, v_2) + R(T(u_1, u_2)cW(v_1, v_2))

= ST(u_1, u_2)W(v_1, v_2) + (RT(u_1, u_2)W(cv_1, cv_2)) = ST(u)W(v) + RT(u)W(cv),

which completes the proof. **[]**

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