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Linear Transformation

Homomorphisms, Kernels and Prime Ideals

Given an onto ring homomorphism &#934;: R1 --> R2 we saw in class that if I <u>C</u> R1 is an ideal containing the kernel of &#934;, then &#934;(I) is an ideal of R2. In addition, we saw that if J <u>C</u> R2 is an ideal, then &#934;&#713;¹(J) is an ideal of R1 containing kernel of &#934;. (i) Show that if I <u>C</u> R1 is a

Job Shop Problem

Suppose you have N jobs that have to be processed on a single machine. For i = 1, 2, . . . ,N, job i requires pi units of time on the machine, and has weight wi. The objective is to schedule these jobs so as to minimize the sum of the weighted completion time of all the jobs, where the completion time of job i is the time at w

Linear Transformations : Polar and Rectangular Coordinates

3. Find the region onto which the half plane y > 0 is mapped by the transformation w = (1+ i)z by using (a) polar coordinates; (b) rectangular coordinates. Sketch the region. Ans. v > u. (This problem is from Linear Transformations.) (Please try to draw the graph clearly. Thank you.)

Transforming Representations

Derive the equation of the line through the points ... and ... in the ... plane, shown in Fig. 37. Then use it to find the linear function ... which can be used in equation (9), Sec. 38. to transform representation (2) in that section into representation (10) there.... The parametric representation used For any given arc C is

Using Excel Solver : Maximizing Interest and Cash Flow

At the start of the year a company wants to invest excess cash in one-month, three-month and six-month CD's. The company is somewhat conservative and wants to make sure it has a safety margin of cash on hand each month. (left over from previous month/ available at the outset, plus principal and interest from CD's that have becom

Topology : Homomorphism (Question B5)

Please see the attached file for the fully formatted problems. B5. (a) Define a homomorphism between topological spaces X and Y. Define what is meant by a topological invariant. (b) State what it means for a map f X -?> Y to be open. Show that a continuous open bijection is a homomorphism. (c) (i) Recall that Fr E, the fron


Tree growth scatterplot. Age 5, 14, 29, 16, 16, 26, 6, 25, 7, 18 Diameter 8, 23, 34, 24, 24, 10, 30, 14, 13 Summary points for first middle and last group based on median-median.(show cords) Equation of the line passing through the summary points. Calculate distance from line through the outer summary points to the middle p

Kernel and Homomorphism

Here's my problem: If A and B are subsets of a group G, define AB = {ab | a 2 A, b 2 B}. Now suppose phi: G -> G0 is a homomorphism of groups and N = Ker(phi) is its kernel. (i) If H is a subgroup of G, show that HN = NH. (Warning: this is an equation of sets; proceed accordingly; do not assume that G is abelian.) (i

Vector Space : Linear Dependence and Null Space

If there are n vectors v1, v2, in E^m, which spans a subspace of dimension k<=n. If k<n, how many different linear dependencies will there be among v1, v2, Can we determine the theorem about the null space and range of a linear transformation about this? (See attachment for full question)


Please assist me with the attached homomorphism questions. Thanks! Example: ? Let f: G -->H be a group homomorphism with kernel K = Ker(f), show that f is one to one if and only if K = ...

Find a self-complementary graph with five vertices.

Let G be a graph. Then G = (V, E), where V and E are the vertex set and edge set, respectively, of G. The complement of G, which we will refer to as "G bar," is the graph (V, E bar), where V is the vertex set of G bar (i.e., the vertex set of G bar is identical to the vertex set of G) and E bar is the edge set of G bar. The e

Prove that graphs that are isomorphic have the same number of vertices and the same number of edges, and that the degree of a vertex of a graph is equal to the degree of the image of that vertex under a graph isomorphism. Also, give an example of a pair of non-isomorphic graphs that have the same number of vertices and the same number of edges.

What does it mean for two graphs to be the same? Let G and H be graphs. We say that G is isomorphic to H provided that there is a bijection f:V(G) -> V(H) so that for all a, b, in V(G) there is an edge connecting a and b (in G) if and only if there is an edge connecting f(a) and f(b) (in H). The function f is called an isomorphi

Kernels, image, nullity and rank.

Find the matrix A of T with respect to the standard basis...of both V and W. Compute the kernel, the image, the nullity and the rank of T. (See attachment for full question)

Kernal and Image

Let V=W= T^3 and let T: V-> W be the projection onto the xy plane sending (x,y,z) to (x,y,o). (a)determine the kernel of T (b)determine the image of T (c)... (See attachment for full question)

Integral Equation

Verify that the solution of u"=f(x), u(0)=0, u(1)=0 given by u(x)= the integration from 0 to 1 of k(x,y)f(y)dy. Use Leibniz rule. (See attachment for full question)

Approximation of Functions:

See PDF attachment! Solve the minimization problem and determine whether there is a unique value of alpha the gives the minimum?

Laplace Transform - Show that the Laplace transform......

Let f be a piecewise continuous function on [0,T]. Define f on the whole of [0,inf) by f(t+nT) for all t and all integer n. Show that the Laplace transform if f is given by L[f(t)] = 1/[1-exp(-sT)]*int(exp(-st)*f(t)dt,t=0..T) By taking the Laplace transform and using the convolution theorem, obtain the solution of

Vectors : Identities and Dot Products

How could you use the properties of the dot product to prove the following identities: (where u and v denote vectors in Rn) a) ||u + v||^2 + ||u-v||^2 = 2(||u||^2 + ||v||^2) b) ||u + v||^2 - ||u-v||^2 = 4u dot v Note: dot = dot product ^ = power ||= distance


Let T be defined on real two dimensional plain, and that: (x,y)T = (ax+by, cx+dy) ; a, b, c, d real constants. Prove that T is a vector space homomorphism. What value of a, b, c, d will T be an isomorphic or isomorphism?


1. Let T be any automorphism of G, show that ZT<(subset) Z. If G is a group and Z is the center of G.

Checking for a linear transformation

This chapter starts as follows rotations about the origin and all reflections in lines through the origin can be expressed as functions with rules of the form x ---> Ax where A is a 2 x 2 matrix any function with such a rule is called a linear transformation a linear transformation of the plane is a function of the form

Linear transformation

f(2,1) = (2,1) Either explain why f is not linear or write down the matrix that represents f . iI general how do you solve problems of this type?

Geometric Interpretation of the Subspace

Show that the set of all elements of R^2 of the form (a, -a), where a is any real number, is a subspace of R^2. Give a geometric interpretation of the subspace.