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# Linear Transformation

### 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

### Homomorphism Proof Functions

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)

### 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

### 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). Write 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 work

### Kernel of Homomorphism

Please see the attached file for the fully formatted problems. Suppose &#61546; is an onto homomorphism from &#8484;16 to a group G of order 4. Find ker(&#61546;). Explain your answer.

### 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 ?

### Extension of A_5

Extension of A_5

### 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!

### Difficulty in setting up the constraints for the problem

The famous Y.S. Chang Restaurant is open 24 hours a day. Waiters and busboys report for duty at 3am, 7am, 11am, 3pm, 7pm or 11pm and each works an 8-hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. The scheduling problem is to determine how many w

### Orthogonal Projection Matrix Theory

See attached file for full problem description with symbols and equations. --- Definition 11.1 An orthogonal projection operator is a linear transformation such that and . Question: If W is a subspace of V, prove that P_w is an orthogonal projection. (P_w is P sub w)

### Apply the Fundamental Homomorphism Theorem.

Homomorphism Problem 3: Define µ : Z4 × Z6 -> Z4 × Z3 by µ ([x]4,[y]6) = ([x+2y]4,[y]3). (a) Show that µ is a well-defined group homomorphism. (b) Find the kernel and image of µ, and apply the fundamental homomorphism theorem.

### 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.

### Group homomorphisms

Homomorphism Problem 1: Find all group homomorphisms from Z4 into Z10

### Linear Transformation to Different Dimensions

Is the following transformation linear? T: R^3 -> R^2 defined by T(x,y,z)=(x,y)