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Linear Transformation

Scatterplots for Tree Growth

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

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)

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)

Area

A rancher wants to fence in an area of 1000000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?

Proving that an Inverse Transformation is a Subspace

Let L: V -- W be a linear transformation, and let T be a subspace of W. The inverse image of T denoted L^-1(T), is defined by L^-1(T) = {v e V | L(v) e T}. Show that L^-1(T) is a subspace of V. A linear transformation L: V -- W is said to be one-to-one if L(v1) = L(v2) implies that v1=v2. Show that L is one-to-one if

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

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)}?

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)

Normals for Linear Transformation

Please see the attached file for full problem description. The "norm" of a vector is a measure of it's "size" or "magnitude". There are many different ways to express the norm of a vector. These are referred to as the Lk norm, and written....see attached

Nilpotent transformation

Consider the transformation N: V->V. Let g be a vector such that N^k-1 does not equal 0, but N^k = 0. First show that the vectors g,N(g),N^2(g),..,N^k-1(g) are linearly independent, and then (assuming V has dimension n) If N is nilpotent of index n, show that the set S= {g, N(g), N^2(g),...,N^n-1(g)}is a basis for V. Describe th

Idempotent linear transformation

A linear transformation L:V->V is said to be idempotent if L dot L = L. If L is idempotent, show that there exists a basis S={a1,a2,...,an} for V such that L(ai)=ai for i= 1,2,...,r and L(aj) = 0v for j= r+1,...,n, where r= p(L). Describe the matrix representing L with respect to the basis S.

Powells Search Method Example

The attached file has some slides provided by my professor on the univariate method and powell's method. I am having trouble understanding, so I tried to work an example, but I am not getting very far. As you work the example, could you explain each step as you go. My professor tried but he and I both ended up frustrated, a

Eigenvalues of a Transformation

The linear operator T: R^3 -> R^3 defined by T(x_1, x_2, x_3) = (x_1 - 3x_3, x_1 + 2x_2 + x_3, x_3 - 3x_1). Find the eigenvalues of the transformation T. Show work. (See attachment for the full question.)

Solving Matrix Representation: Linear Transformation

** Please see the attached file for full problem description ** Let T be a linear operator on P_3 defined as follows: T(ax^3 + bx^2 + cx + d) = (a - b)x^2 + (c - d)x + (a + b - c). The matrix [T]_G which represents T with respect to the basis G which = {1 + x, 1 - x, 1 - x^2, 1 - x^3}. Show that

Irreducible Polynomial : Galois Group

4. Find an irreducible polynomial defining the field extension K = Q (cube root 2, sq root &#8722; 3) over Q . Is K a normal extension of Q ? What is the Galois group for the splitting field of the polynomial defining K over Q ?

Linear Algebra -- Linear Transformations

Let a be a fixed vector in R2. A mapping of the form L(x) = x+a is called a translation. Show that if a does not equal 0, then L is not a linear transformation. Describe or illustrate geometrically the effect of the translation. Thanks for your help!

Homomorphisms

Homomorphism (a) Find the formulas for all group homomorphisms from Z_18 into Z_30. (b) Choose one of the nonzero formulas in part (a), and for this formula find the kernel and image, and show how elements of the image correspond to costs of the kernel.