Problem 4. Consider ϕ : (Z100, +100) → (Z100, +100) defined by ϕ([x]100) = [41x +...

Problem 4. Consider ϕ : (Z100, +100) → (Z100, +100) defined by ϕ([x]100) = [41x + 19]100. a.) Show that ϕ is well-defined. b.) Use the fact that 61 · 41 = 2501 to argue that the function ϕ is one-to-one. c.) Is ϕ onto? Why or why not. Suppose that y is in the image of ϕ, find an element x so that ϕ(x) = y. [The answer will be dependent on the value of y.] Then prove that the found element works. d.) Is ϕ a homomorphism?

Expert Solution

Colby Messinger answered 2 years ago

Consider the ring homomorphism ϕ : Z[x] →R defined by ϕ(x) = √5. Let I =...

Consider the ring homomorphism ϕ : Z[x] →R defined by ϕ(x) = √5. Let I = {f ∈Z[x]|ϕ(f) = 0}. First prove that I is an ideal in Z[x]. Then find g ∈ Z[x] such that I = (g). [You do not need to prove the last equality.]

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x...

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x +2y +z, 2x +2y +3z +w, x +4y +2w) a) Find the dimension and basis for Im T (the image of T) b) Find the dimension and basis for Ker ( the Kernel of T) c) Does the vector v= (2,3,5) belong to Im T? Justify the answer. d) Does the vector v= (12,-3,-6,0) belong to Ker? Justify the answer.

Problem 4 (Sets defined inductively) [30 marks] Consider the set S of strings over the alphabet...

Problem 4 (Sets defined inductively) [30 marks] Consider the set S of strings over the alphabet {a, b} defined inductively as follows: • Base case: the empty word λ and the word a belong to S • Inductive rule: if ω is a string of S then both ω b and ω b a belong to S as well. 1. Prove that if a string ω belongs to S, then ω does not have two or more consecutive a’s. 2....

4. Consider the random variable Z from problem 1, and the random variable X from problem...

4. Consider the random variable Z from problem 1, and the random variable X from problem 2. Also let f(X,Z)represent the joint probability distribution of X and Z. f is defined as follows: f(1,-2) = 1/6 f(2,-2) = 2/15 f(3,-2) = 0 f(4,-2) = 0 f(5,-2) = 0 f(6,-2) = 0 f(1,3) = 0 f(2,3) = 1/30 f(3,3) = 1/6 f(4,3) = 0 f(5,3) = 0 f(6,3) = 0 f(1,5) = 0 f(2,5) = 0 f(3,5) = 0 f(4,5) = 1/6...

Consider the graph defined by f(x) = (x-3)2 a) state the coordinates of the vertex and...

Consider the graph defined by f(x) = (x-3)2 a) state the coordinates of the vertex and the direction of opening of f(x) b) Find the maximum and minimum values of f(x) on the interval 3<=x<=6 c) Explain how you could find part b) without finding the derivative

consider a world off to good x and y and consider the following preferences defined over...

consider a world off to good x and y and consider the following preferences defined over all bundles of positive quantities of these two goods (Xa,Ya)>=(Xb,Yb) if and only if Xa>Xb or Xa=Xb and Ya>=Yb in other words this consumer always prefers the bundle that has more x. If two bundles have the same amount of X, then the consumer rephrase the one with more y 1) show that this preference violates one of the axioms of rational choice

(a) Apply the chain rule to express ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ using ∂/∂x, ∂/∂y, and ∂/∂z....

(a) Apply the chain rule to express ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ using ∂/∂x, ∂/∂y, and ∂/∂z. (b) Solve algebraically for ∂/∂x, ∂/∂y, and ∂/∂z with ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ when ρ does NOT equal 0 and sin ϕ does NOT equal 0. (Hint: you can use method of elimination to reduce the number of variables.) (c) Express ∂2/∂x2 with ρ, ϕ, θ, and their partials.

Consider the initial value problem X′=AX, X(0)=[−4,-2], with A=[−6,0,1,−6] and X=[x(t)y(t)] (a) Find the eigenvalue λ,...

Consider the initial value problem X′=AX, X(0)=[−4,-2], with A=[−6,0,1,−6] and X=[x(t)y(t)] (a) Find the eigenvalue λ, an eigenvector V1, and a generalized eigenvector V2 for the coefficient matrix of this linear system. λ= , V1= ⎡⎣⎢⎢ ⎤⎦⎥⎥ , V2= ⎡⎣⎢⎢ ⎤⎦⎥⎥ $ (b) Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers. X(t)=c1 ⎡⎣⎢⎢ ⎤⎦⎥⎥ + c2 ⎡⎣⎢⎢ ⎤⎦⎥⎥ (c) Solve the original initial value problem. x(t)=...

Consider the Hamiltonian of a particle in one-dimensional problem defined by: H = 1 2m P...

Consider the Hamiltonian of a particle in one-dimensional problem defined by: H = 1 2m P 2 + V (X) where X and P are the position and linear momentum operators, and they satisfy the commutation relation: [X, P] = i¯h The eigenvectors of H are denoted by |φn >; where n is a discrete index H|φn >= En|φn > (a) Show that < φn|P|φm >= α < φn|X|φm > and find α. Hint: Consider the commutator [X, H] (b)...

Consider rolling two dice and let (X, Y) be the random variable pair defined such that...

Consider rolling two dice and let (X, Y) be the random variable pair defined such that X is the sum of the rolls and Y is the maximum of the rolls. Find the following: (1) E[X/Y] (2) P(X > Y ) (3) P(X = 7) (4) P(Y ≤ 4) (5) P(X = 7, Y = 4)

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