4. Prove that the function that sends an ordered pair (p,q) of elements in P(R) to * p(x) q(x)e da defines an inner product on P(R). 0

Q: where did the 1.18 come in how did you get that

A:

Q: 1. Factorize each of the following polynomials over Z5 [x]. a) f(x) = x³ + 3x² + 3x + 2 b) f(x) = x¹…

A: The important thing to observe is that we have to check polynomials modulo 5 while factoring over…

Q: 6: F7 → F2 where (x1, x2,...,x7) = (X1 + X2, X3 + x7). Prove is not a linear functional on F7

A:

Q: A relation < is defined on R? by (x1, x2) < (y1, y2) if and only if x1 < yi and x1 + x2 2 yı + y2.…

A:

Q: 4. Consider the set of functions S = {-3(x² – 4)°,7|x² – 4|³}. a. Show that S is a linearly…

A:

Q: Let f: IR → R and g: R → R with f(x) = 5x – 7 and g(x) = x2 – 4x + 4 Find: a) Domain of . g(D(f.g))…

A:

Q: Acetaminophen is a common pain reliever and fever reducer Here is a graph showing the amount of…

A: a. From the graph, it is seen that the vertical intercept is at (0,400) It implies that initially an…

Q: Let T: P2 → P2 be defined by T(p(x)) = xp'() A. Describe the kernel of T. B. Describe the range of…

A:

Q: Determine all homomorphisms from D5 onto Z₂ Z₂. Determine all homomorphisms from D5 to Z₂ + Z₂.

A: We have to determine all homomorphism from  onto .We also have to find homomorphism from  to

Q: a) Let f: Z Z be such that f(x) 2*x. Is f invertible, and if so, what is its inverse?

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: Let f(T) and g(T) be polynomials in Q[T] whose gcd in Q[T] is 1. Show that 1 is also their gcd in…

A:

Q: (5) Let ƒ : A → B and g : B → C, where f and g are functions and A, B, and C are sets. (a) Prove…

A: In 2 nd part construct a function

Q: Let Ps have the inner product given by evaluation at-2,-1, 0,1, and 2. Let p(t) = t2- 2t and q(t) =…

A: Let P3 have the inner product given by evaluation at -2, -1, 0, 1 and 2 . Let p(t) = t2-2t  and q(t)…

Q: Suppose that β is a zero of f (x) =x4 + x + 1 in some extensionfield E of Z2. Write f (x) as a…

A: Given that β is the zero of fx=x4+x+1 in some extension field E of Z2. We have to write fx as a…

Q: Let x0, x1, ..., Xn be distinct points on the interval [a, b], where Xo <x₂ < x₂ <--- < x₂ and let f…

A:

Q: Let f(x)= x3 + x2 + x +1  ∈ Z2[x]. Write f(x) as a product of irreduciblepolynomials over Z2.

A: Given polynomial is: f(x)=x3+x2+x+1 Since, f(1)=0 So, x-1 is a factor. Now, factorize the…

Q: Prove that the function ƒ : R – {5} —> R - {3} defined by f(x) = 3x +1 x-5 is a bijection.

A: Consider the function  defined by .To prove that f is a bijection, we have to prove that f is both…

Q: Show that the function defines an inner product for polynomials p(x) = a0 + a1x + . . . + anxn and…

A:

Q: Let L be a linear Prove that dim[Range (L)] + dim[Kernel (L)] = dim[V] transformation from V W

A:

Q: 2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +)…

A:

Q: Show that U(8) is not isomorphic to U(10).

A: To Show: U(8) is not isomorphic to U(10)

Q: The function f : R → R defined by the formula f(x) = 3x − 5 for all x ∈ R was shown to be one-to-one…

A:

Q: 3. Determine whether each of the following functions is a bijection and explain why. (a) f: R R,…

A: Note:As our company  guidelines we are supposed to answer only first 3 sub-parts. kindly re-post…

Q: Let V be the set of all pairs (x, y) of real numbers together with the following operations: (x₁,…

A: V be the set of all pairs (x,y) of real numbers  together with the following operations ;…

Q: Prove that the mapping f : U(16) ---> U(16) given by f(x) = x^5 is an automorphism.

A: The given map is f:U16→U16 defined by fx=x5. We have to prove that f is an automorphism. We know…

Q: Show that (p(x), g(x)) = p(0)q(0) +p(1)q(1) + p(2)q(2) defines an inner product on P2.

A:

Q: prove V injective if x1 ‡xn → v(x1) = v(xn)

A:

Q: f: R →R by f(x)= x^2 and V is a subset of the codomain. Find a set V such that V has only one…

A: We are given that f : R → R by f(x) = x2 and V is a subset of the codomain. We need to find the set…

Q: . Let a := V1+ v3. Find the minimal polynomial of a over Q.

A:

Q: 4) Let ƒz = dƒ/əz and ƒz = 0ƒ/əz. Recall ƒ = u + iv, the notation f is used for the complex…

A:

Q: 21. Show that the set of polynomials {C₁, C₂ (2x - 1), C3 (6x² − 6x + 1)} in P(R) is orthogonal with…

A:

Q: (b) Let the function g : Z → Z be defined by g(x) = 7x + 3. (i) Show that g is one-to-one. (ii) Show…

A: