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# Linear Transformation

### Topology : Homomorphism Formatted

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

### Transformation Vector Spaces

Please see Attachment Respond in PDF format

### 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

### Automorphisms Group

Let G=Zp+Zp .How many automorphisms does G have? Please explain clearly the counting principle.

### Homomorphism of cyclic group

Let G be a group and h:G-->h(G) a homomorphism.Prove or Disprove If G is cyclic,then h(G)is cyclic.

### 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

### Transformation : Null Space, Range and Eigenvalues

Please see the attached problem relating to null space, range, eigenvalues and transformation.

### 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)

### How to Solve a Minimization Problem

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

### 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

### Laplace Transform of a Periodic Function

See the attached file. 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,

### 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.

### Vector Space 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.

### Transformations in Hom(v,v)

Prove that the invertible transformations in Hom(v,v) form a group under multiplication

### 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

### How to Explain Why a Transformation is Not Linear

f(2,1) = (2,1) Either explain why f is not linear or write down the matrix that represents f . In 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: G =~ G1 means G is isomorphic to G1 If G/K =~ H, show that there exists an onto homomorphism \$:G -> H with kernel \$ = K

### Groups : Isomorphism and Homomorphism

Show that a group G is simple if and only if every nontrivial group homomorphism G -> G1 is one-to-one.

### 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)}?

### Lower triangular decomposition

Problem attached.

### Automorphism Normal Subgroup

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