Proving Properties of Compositions of Linear Transformations

Posted by: tyr 6 years, 10 months ago

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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

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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. []

• Tags:
• Linear Algebra
• algebra
• vectors
• proof
• real
• linear transformations

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