PLEASE write it on PAPER. 17. Show that if g is a function in C (a, b), then Tg(x) isa linear transformation from C(a, b) to C (a, b). Proof We will prove the associativity property in part (6) and leave the proofs of the remainingparts as exercises (Exercise 16). In fact, this associativity property holds for all functions,not just linear transformations. To prove it (which is the approach used to prove all ofthese properties), we take a vector v in the domain and verify that the functions on eachside of the equation applied to v give us the same result: Applying the definition of thecomposite of functions, we haveR(ST)(v) = R(ST (v)) = R(S(T(v)))and(RS)T(v) = RS(T(v)) = R(S(T(v))).Therefore, the functions R(ST) and (RS)T are the same.UTOLEOne other similarity between matrices and linear transformations we mention at thistime involves exponents. Recall that if A is a square matrix and n is a positive integer,we have defined the nth power of A, A", as the product of n factors of A. Notice thatwe can do likewise for a linear transformation that has the same domain and codomain.That is, if T: V → V is a linear transformation and n is a positive integer, then 7" isthe product (composite) of n factors of T.We are now going to see how we can use the ideas we have developed involvinglinear transformations to obtain a particularly elegant approach to the study of lineardifferential equations. To simplify our presentation, let us assume that our functions havederivatives of all orders on an open interval (a, b); that is, we will use C (a, b) as ourvector space of functions. We will make use of two basic types of linear transformations.One type involves the differential operator D C (a, b) → C (a, b) and its powersD", which amount to taking the nth derivative. The other type involves multiplyingfunctions f in C (a, b) by a fixed function g in C (a, b): If g is a function in C (a, b),definebyFor example, if g(x) = x²,Tg(x): C (a, b) → Co (a, b)Tgix) (f(x)) = g(x)f(x).Tg(x) (f(x)) = T₂²(f(x)) = x² f(x).We leave it to you to verify that Tg(x) is a linear transformation (Exercise 17).

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PLEASE write it on PAPER. 17. Show that if g is a function in C (a, b), then Tg(x) isa linear transformation from C(a, b) to C (a, b). Proof We will prove the associativity property in part (6) and leave the proofs of the remainingparts as exercises (Exercise 16). In fact, this associativity property holds for all functions,not just linear transformations. To prove it (which is the approach used to prove all ofthese properties), we take a vector v in the domain and verify that the functions on eachside of the equation applied to v give us the same result: Applying the definition of thecomposite of functions, we haveR(ST)(v) = R(ST (v)) = R(S(T(v)))and(RS)T(v) = RS(T(v)) = R(S(T(v))).Therefore, the functions R(ST) and (RS)T are the same.UTOLEOne other similarity between matrices and linear transformations we mention at thistime involves exponents. Recall that if A is a square matrix and n is a positive integer,we have defined the nth power of A, A&#34;, as the product of n factors of A. Notice thatwe can do likewise for a linear transformation that has the same domain and codomain.That is, if T: V → V is a linear transformation and n is a positive integer, then 7&#34; isthe product (composite) of n factors of T.We are now going to see how we can use the ideas we have developed involvinglinear transformations to obtain a particularly elegant approach to the study of lineardifferential equations. To simplify our presentation, let us assume that our functions havederivatives of all orders on an open interval (a, b); that is, we will use C (a, b) as ourvector space of functions. We will make use of two basic types of linear transformations.One type involves the differential operator D C (a, b) → C (a, b) and its powersD&#34;, which amount to taking the nth derivative. The other type involves multiplyingfunctions f in C (a, b) by a fixed function g in C (a, b): If g is a function in C (a, b),definebyFor example, if g(x) = x²,Tg(x): C (a, b) → Co (a, b)Tgix) (f(x)) = g(x)f(x).Tg(x) (f(x)) = T₂²(f(x)) = x² f(x).We leave it to you to verify that Tg(x) is a linear transformation (Exercise 17).

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17. Show that if g is a function in C transformation from C a linear (a, b), then Tg(x) is (a, b) to C (a, b).