Problem number 2 please. Let f(x) be a polynomial of degree n in P_n(Ropf). Prov

Question

Problem number 2 please.

Let f(x) be a polynomial of degree n in P_n(Ropf). Prove that for any g Element P_n(Ropf) there exist unique scalars c_0, c_1, TripleDot, C_n such that g(x) = c_0f(x) + c_1f'(x) + c_2f"(x) + TripleDot + C_nf^(n)(x). Let V be a vector space over a field F and let V_1 and V_2 be finite-dimensional subspaces of V such that V_1 + V_2 : = {v_1 + v_2|v_1 Element V_1, v_2 Element V_2} is equal to V. Prove that dim(V) lessthanorequalto dim(V_1) + dim(V_2), and that equality holds if and only if V_1 Intersection V_2 = {0}. Let V and W be vector spaces, not necessarily finite-dimensional. Let beta be a basis for V. Prove that if T is 1-1 then T(beta) is a linearly independent set of vectors in W. Prove that if T is onto then T(beta) spans W.

Explanation / Answer

Solution :- (1)

Let f(x) be a polynomial of degree n in Pn(R).

We know that the definition of the linearly indepndence of the vectors.

The vectors in set S = {v1,v2,...,vn} are said to be linearly independent, if there exist a scalars a1,a2,...,an. not all zero such that a1v1 +a2v2+...+anvn=0

Since f(x) is the nth degree polynomial. The set { f , f ' , f '' , ...f(n) } is linearly independent .

So the set { f , f ' , f '' , ...f(n) } is a basis for the Pn(R).

Since g(x) Pn(R) , by the definition of basis ,

it must be a linear combination of the elements of the basis { f , f ' , f '' , ...f(n) }

Therefore  g(x) = c0f(x) + c1f '(x) + c2f "(x) +...+ cnf (n)(x).

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.