Suppose A is the set of positive real numbers, and suppose u and v are two...

Suppose A is the set of positive real numbers, and suppose u and v are two strictly increasing functions.1 It is intuitive that u and v are ordinally equivalent, since both rank larger numbers higher, and therefore generate the same ranking of numbers. Write this intuition as a proof.

Expert Solution

Colby Messinger answered 1 year ago

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 +...

(1) Suppose that V is a vector space and that S = {u,v} is a set...

(1) Suppose that V is a vector space and that S = {u,v} is a set of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so thatT is another set of two vectors in V ). (a) Show that if S is linearly independent in V then T is also independent. (Hint: suppose that there is a linear combination of elements of T that is equal to 0. Then ....). (b) Show that if S generates V...

Find five positive natural numbers u, v, w, x, y such that there is no subset...

Find five positive natural numbers u, v, w, x, y such that there is no subset with a sum divisible by 5

Let V be the set of all ordered pairs of real numbers. Consider the following addition...

Let V be the set of all ordered pairs of real numbers. Consider the following addition and scalar multiplication operations V. Let u = (u1, u2) and v = (v1, v2). Show that V is not a vector space. • u ⊕ v = (u1 + v1 + 1, u2 + v2 + 1 ) • ku = (ku1 + k − 1, ku2 + k − 1) 1)Show that the zero vector is 0 = (−1, −1). 2)Find the...

Consider the set of real numbers S=[-1,0)U{1/n:n=1,2,3,...}. Give the explicit set for each of the The...

Consider the set of real numbers S=[-1,0)U{1/n:n=1,2,3,...}. Give the explicit set for each of the The complement of S, The closure of S, The boundary points of S, The limit points of S, The isolated points of S, The interior of S, and The exterior of S.

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ =...

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4, ∥u + v∥ = 5. Find the inner product 〈u, v〉. Suppose {a1, · · · ak} are orthonormal vectors in R m. Show that {a1, · · · ak} is a linearly independent set.

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.

using theorem 11.10 (First Isomorphism Theorem), Show that the set of positive real numbers with multiplication...

using theorem 11.10 (First Isomorphism Theorem), Show that the set of positive real numbers with multiplication is isomorphic to the set of real numbers with addition. Theorem 11.10 First Isomorphism Theorem. If ψ : G → H is a group homomorphism with K = kerψ, then K is normal in G. Let ϕ : G → G/K be the canonical homomorphism. Then there exists a unique isomorphism η : G/K → ψ(G) such that ψ = ηϕ.

Let V be the set of positive reals, V = {x ∈ R : x >...

Let V be the set of positive reals, V = {x ∈ R : x > 0}. Define “addition” on V by x“ + ”y = xy, and for α ∈ R, define “scalar multiplication” on V by “αx” = x^α . Is V a vector space with these unusual operations of addition and scalar multiplication? Prove your answer.

Suppose we define a relation on the set of natural numbers as follows. Two numbers are...

Suppose we define a relation on the set of natural numbers as follows. Two numbers are related iff they leave the same remainder when divided by 5. Is it an equivalence relation? If yes, prove it and write the equivalence classes. If no, give formal justification.

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