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Linear Transformation

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

Scatterplots

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, v3...vn 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, v3...vn? Can we determine the theorem about the null space and range of a linear transformation about this? (See attachment for full question)

Homomorphisms

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

Isomorphism

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?

Automorphism

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.

Groups : Isomorphism and Homomorphism

Note: S4 means symmetric group of degree 4 A4 means alternating group of degree 4 e is the identity Is there a group homomorphism $:S4 -> A4, with kernel $ = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}?

Automorphism

Show that innG is the normal subgroup of autG for any group G Note: innG = inner automorphism group of G aut G = automorphism group of G

Homomorphisms

If G = <X> and $:G->G1 is an onto homomorphism, show that G1 = <$(X)>, where $(X) = the set of $(x) given that x belongs to X.

Isomorphism

Note: ~~ means an isomorphism exists. Moreover,if an isomorphism existed from G to G1 I would say G ~~ G1 Questions: If G is an infinite cyclic group, show that G ~~ Z (Z is the set of integers)