Explore BrainMass

Linear Transformation

Excel Solver - maximize the annual passenger-carrying capability.

An airline owns an aging fleet of jet airplanes. It is considering a major purchase of up to 17 new model 7a7 and 7b7 jets. The decision must take into account numerous cost and capability factors, including the following: (1) The airline can finance up to $400 million in purchases; (2) each 7a7 jet will cost $35 million and eac

Poisson Kernel and Harmonic Function

Let P_r(t)=R((1+z)/(1-z)), z=re^it be the Poisson kernel for the unit disc |z|<1. Let U(theta) be a continous function of the interval [0,pi] with U(0)=U(pi)=0. Show that the function u(re^itheta)=1/2pi(integral from 0 to pi of {P_r(t-theta)-P_r(t+theta)}U(t)dt is harmonic in the half-disc {re^itheta,0<=r<1, 0<=theta<=pi} and

Homomorphism and First Isomorphism Theorem

Let R>0 be the group of positive real numbers under multiplication. Let CX be the group of nonzero complex numbers under mu!tiplication. Let S1 = {a + bi such that a^2 + b^2 = 1) be the subgroup of C consisting of all complex numbers of absolute value 1. Note that is normal in Cx since Cx is abelian. Prove that CX/S1 is isomorph


Let G be a group and let H be a normal subgroup of G. Let m be the index of H in G (that is, the number of cosets of H). Prove that for any a we have am H. (b) Give an example of group G, a subgroup H of index in, and an element a G such that am is not in H. (Of course, your subgroup H had better not be normal.) (4) (a) Suppos

Hamilton Quaternions : Kernels and Images

Show that the map phi: H->M_2(C) defined by phi(a+bi+cj+dk)=(a+b(sqrt -1),c+d(sqrt-1);-c+d(sqrt-1),a-b(sqrt-1)) is a ring homomorphism. Calculate its kernel and describe its image. Ps. Here H is the ring of integral Hamilton Quaternions and M_2(C) are 2x2 matrices with complex coefficients. notation after the equal sign in

Homomorphisms and Surjections

Let f:G->H be a group homomorphism. Prove or disprove the following statement. 1.Let a be an element of G. If f(a) is of finite order, then a is also of finite order. 2.Let f be a surjection. Then f is an isomorphism iff the order of the element f(a) is equal to the order of the element a , for all a belong to G. F

Vector Spaces, Basis and Quotient Spaces

1. Let and be vector spaces over and let be a subspace of Show that for all is a subspace of and this subspace is isomorphic to . Deduce that if and are finite dimensional, then dim = (dim - dim )dim 2. Let be a linear operator on a finite dimensional vector space Prove that there is a basis

Normal subgroups, Second Theorem of Isomorphism, Conjugates and Cyclic Groups

Problem 1. Let a,b be elements of a group G Show a) the conjugate of the product of a and b is the product of the conjugate of a and the conjugate of b b) show that the conjugate of a^-1 is the inverse of the conjugate of a c)let N=(S) for some subset S of G. Prove that the N is a normal subgroup of G if gSg^-1<=N for

Linear Transformations and Subspaces

B1) This question concerns the following two subsets of : (a) Show that , and find a vector in that does not belong to T. [3] (b) Show that T is a subspace of . [4] (c) Show that S is a basis for T, and write down the dimension of T. [7] (d) Find an orthogonal basis for T that contains the vector .

Linear Combinations, Basis and Transformations

1. Given a basis B = { u1 = [1, 2], u2 = [2, 1] } for R^2, express u = [7, -2] as a linear combination of u1 and u2. How many ways can you do this? (in this problem...the u1 and u2 should actually be u sub 1 and u sub 2...I couldn't do that notation here....also the u1, u2, and u should all be bold to represent vectors) 2. L


This week lecture is taught about Isomorphism, automorphism and Inner automorphism, but I don't understand what they are. Can you give some simple examples?


Please help. I only need answers with brief explanations. No need of detailed working. (See attached file for full problem description) --- State whether the following are true or false with reasons: 1. If a in S6, then an =1 for some n greater than or equal to 1. 2. If axa-1=bxb-1, then a=b 3. The function e


We are working on the proof of showing G (the group of rigid motions of a regular dodecahedron) is isomorphic to the alternating group A_5. Lemma: Let H be a normal subgroup of a finite group G, and let x be an element of G. If o(x) and [G:H] are relatively prime, then x is in H. Theorem: Any 60 element group having 24 el

Linear Mapping, Linear Space, Differentiability and Continuity

In each of Exercises 40 through 46 following, a linear space V is given and a mapping T : V&#8594;V is defined as indicated. In each case determine whether T is a linear mapping. If T is linear, determine the kernel (or null space) and range, and compute the dimension of each of these subspaces wherever they are finite-dimension

Laplace Transformation - Initial Value

What is the best statement that you can make about the existence and uniqueness of the solution of the following initial value problems? (a) y'= sin(ty)+1/t, y(1)=2 (See attachment for full question)

Linear Transformations, Change of Basis and Conjugation

Let V = Q3 and let ' be the linear transformation from V to itself: '(x, y, z) = (9x + 4y + 5z,&#8722;4x &#8722; 3z,&#8722;6x &#8722; 4y &#8722; 2z), x, y, z E Q With respect to the standard basis B find the matrix representing this linear transformation. Take the basis E = {(2,&#8722;1,&#8722;2), (1, 0,&#8722;1), (3,&#8722;2