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Linear Transformation

Linear transformations

See attached file. 1. For each of the following functions decide whether or not the function is a linear transformation.

Minimal polynomial for a linear transformation

Any explanations of the attached definitions is greatly appreciated. Thank you. The definition of the minimal polynomial for the linear transformation.... the relationship between the cosets in G of the stabilizer....

Fields, homomorphisms

Not sure why, but I am having trouble with this one. I'd really appreciate it if someone is willing to help here. Please see the attached file. Thank you

Linear Programming Model for Maximizing Profit of a Production Schedule

Ms. Olsen, a coffee processor, markets three blends of coffee. They are Brand X, Minim and Taster's Reject. Ms. Olsen uses two types of coffee beans, Columbian and Mexican, in her coffee. The following chart lists the compositions of the blends. Blend Columbian Beans Mexican Beans Brand X 80% 20% Minim 50

Linear Transformations, Hilbert Space, Inner Product and Matrix Adjoint

Please see the attached file for the fully formatted problems. Find a 5x5 matrix M>0 such that if and x(t)= then Can we use this definition to find the adjoint of T (T is given at the end)? This part is the additional information to solve the question above; Let V=L2[-1,1] be the Hilbert space of func

Linear Transformations : Hilbert Space and Inner Product

Please see the attached file for the fully formatted problems. Let V=L2[-1,1] be the Hilbert space of functions over the time interval [-1,1] with inner product Let P5 V be the subspace of polynomials of order 4 or less, endowed with the inner product and norm of V, and let , be its natural basis. Define a linear transf

Transformation of State Variables and State-Space Representations

Consider the system . It can be shown that there is other state equation possible for this system. Let us define a new set of state-variables z1 , z2 , z3 by the transformation: Obtain the state-space representation of the system in terms of the new variables. Please see the attached file for the fully formatted problem

Center of Mass of a 3D Pyramid

I need three different algebraic solutions of the center of mass of a square-based pyramid. Finding the center of mass of a square-based 3D pyramid is usually done in calculus, but our professor wants us to work it out in algebra. One hint he gave us (for one way of approaching this problem) is to split the pyramid into small bl

Linear Spaces, Mappings and Dimensional Spaces

1) Show that if dim X = 1 and T belongs to L(X,X), there exists k in K st Tx=kx for all x in X. 2) Let U and V be finite dimensional linear spaces and S belong to L(V,W), T belong to L(U,V). Show that the dimension of the null space of ST is less than or equal to the sum of the dimensions of the null spaces of S and T. 3)

Linear Mappings, Differentiation and Linear Spaces

1) Show that this mapping is linear: T: P5 -> P8 defined as Tp(t)=p(t+1)-p(t)+integral(t-1 to t) s^2 p(s) ds 2) Prove the following is true, or give a counterexample: If l is a nonzero scalar linear function on linear space X (which may be finite or infinite) and a is an arbitrary scalar, there exists a vector x in X st l(x

Economics : Equations, Quantity and Profit

A company manufacturers and sells a product. The estimated demand and cost functions are as below: Demand P = 16000 -2Q(squared) 0< Q < 85 Total cost 1000q = 100000 Where p is unit price (in £'s) q is quantity, tc is total cost (in £'s) A) Find the equation for total revenue hence an equation for profit. B) Find t

Solve: Inventory Management

1. Cooper Automotive Products manufactures components used in the automotive industry. The company purchases parts for use in its manufacturing operation from a variety of different suppliers. One supplier provides a part where the assumptions of the EOQ model are realistic. The annual demand is 5000 units, the ordering cost i

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