Example 1.8. Fix a domain D, and let V be the set of all functions f(t)...

Example 1.8. Fix a domain D, and let V be the set of all functions f(t) defined
on D. Define addition and scalar multiples as with polynomials for all t ∈ D:

(f + g)(t) = f(t) + g(t)
(cf)(t) = cf(t)

Then V is a vector space as well, the axioms are verified similarly to those for Pn.

Verify that V in the previous example satisfies the axioms for a vector space.

Expert Solution

Colby Messinger answered 1 year ago

2. Let A = {1,2,3,4}. Let F be the set of all functions from A to...

2. Let A = {1,2,3,4}. Let F be the set of all functions from A to A. Recall that IA ∈ F is the identity function on A given by IA(x) = x for all x ∈ A. Consider the function E : F → A given by E(f) = f(1) for all f ∈ F. (a) Is the function E one-to-one? Prove your answer. (b) Is the function E onto? Prove your answer. (c) How many functions f ∈...

Let V be the vector space of all functions f : R → R. Consider the...

Let V be the vector space of all functions f : R → R. Consider the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The function T : W → W given by taking the derivative is a linear transformation a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the matrix for T relative to B. b)Find all the eigenvalues of the matrix you found in the previous part and describe their eigenvectors. (One...

3. Let X = {1, 2, 3, 4}. Let F be the set of all functions...

3. Let X = {1, 2, 3, 4}. Let F be the set of all functions from X to X. For any relation R on X, define a relation S on F by: for all f, g ∈ F, f S g if and only if there exists x ∈ X so that f(x)Rg(x). For each of the following statements, prove or disprove the statement. (a) For all relations R on X, if R is reflexive then S is reflexive....

Let U and V be vector spaces, and let L(V,U) be the set of all linear...

Let U and V be vector spaces, and let L(V,U) be the set of all linear transformations from V to U. Let T_1 and T_2 be in L(V,U),v be in V, and x a real number. Define vector addition in L(V,U) by (T_1+T_2)(v)=T_1(v)+T_2(v) , and define scalar multiplication of linear maps as (xT)(v)=xT(v). Show that under these operations, L(V,U) is a vector space.

Let Vand W be vector spaces over F, and let B( V, W) be the set...

Let Vand W be vector spaces over F, and let B( V, W) be the set of all bilinear forms f: V x W ~ F. Show that B( V, W) is a subspace of the vector space of functions 31'( V x W). Prove that the dual space B( V, W)* satisfies the definition of tensor product, with respect to the bilinear mapping b: V x W -> B( V, W)* defined by b(v, w)(f) =f(v, w), f E...

Let T∈ L(V), and let p ∈ P(F) be a polynomial. Show that if p(λ) is...

Let T∈ L(V), and let p ∈ P(F) be a polynomial. Show that if p(λ) is an eigenvalue of p(T), then λ is an eigenvalue of T. Under the additional assumption that V is a complex vector space, and conclude that {μ | λ an eigenvalue of p(T)} = {p(λ) | λan eigenvalue of T}.

Let Ω be any set and let F be the collection of all subsets of Ω...

Let Ω be any set and let F be the collection of all subsets of Ω that are either countable or have a countable complement. (Recall that a set is countable if it is either finite or can be placed in one-to-one correspondence with the natural numbers N = {1, 2, . . .}.) (a) Show that F is a σ-algebra. (b) Show that the set function given by μ(E)= 0 if E is countable ; μ(E) = ∞ otherwise...

let D be an integral domain. prove that an element of D[x] is a unit if...

let D be an integral domain. prove that an element of D[x] is a unit if an only if it is a unit in D.

Let f be a function with domain the reals and range the reals. Assume that f...

Let f be a function with domain the reals and range the reals. Assume that f has a local minimum at each point x in its domain. (This means that, for each x ∈ R, there is an E = Ex > 0 such that, whenever | x−t |< E then f(x) ≤ f(t).) Do not assume that f is differentiable, or continuous, or anything nice like that. Prove that the image of f is countable. (Hint: When I solved...

Let V be the set of all ordered triples of real numbers. For u = (u1,...

Let V be the set of all ordered triples of real numbers. For u = (u1, u2, u3) and v = (v1, v2, v3), we define the following operations of addition and scalar multiplication on V : u + v = (u1 + v1, u2 + v2 − 1, u3 + v3 − 2) and ku = (ku1, ku2, ku3). For example, if u = (1, 0, 3), v = (2, 1, 1), and k = 2 then u +...

Question