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Linear Transformation

Initial value existence and uniqueness

What is the best statement that you can make about the existence and uniqueness of the solution of the following initial value problems? (a) y'= sin(ty)+1/t, y(1)=2 (See attachment for full question)

Linear Transformations, Change of Basis and Conjugation

Let V = Q3 and let ' be the linear transformation from V to itself: '(x, y, z) = (9x + 4y + 5z,−4x − 3z,−6x − 4y − 2z), x, y, z E Q With respect to the standard basis B find the matrix representing this linear transformation. Take the basis E = {(2,−1,−2), (1, 0,−1), (3,−2

Linear Transformations : Basis, Kernel, Image, Onto, One-to-one and Matrices

1) Define a linear transformation.... a) Find a basis for Ker T. b) Find a basis for Im T. c) Is T an onto map? d) Is T a one-to-one map? 2) Define a linear transformation... a) Find the matrix for T with respect to the standard basis. b) Find the matrix for T with respect to { ( ) , ( ) , ( ) } as the basis for R and t

Transformations

I need help and an explanation for the following: Using the matrix A = 1 -1 0 0 -1 1 -1 2 -1 to compute TsubA(x), for x = (1, 2, 3)^T. Here TsubA: R^3 into R^3 is defined by TsubA(x) = Ax. Also describe the kernel of the transformation TsubA (that is state what a typical vector in ker T looks like).

Order of Elements, Factor Groups and Homomorphisms

Help with this linear transformation problem. Please help with the following problem. Provide step by step calculations. A. What is the order of the element 14 + <8> in the factor group Z24 / <8>? I know that if I let G = Z24 and H = <8> , then H = <8> = {0,8,16}. So Z24/<8> = {0+H,1+H,2+H,3+H,4+H,5+H,6+H,7+H}.

Isomorphism of Binary Structures

Determine whether the mapping phi: Z -> Z which is defined by phi(n) = n + 1 is an isomorphism from the binary structure (Z, +) to the binary structure (Z, +). If not, explain why and give a counter-example. See the attached file.

Identity Element Preserved in a Homomorphism

Let ... be a homomorphism of a group ... into a group... . Show that if is the identity element of... , then is the identity element... in ... . Please see the attached file for the fully formatted problems.

Mapping : Homomorphism

Prove that mapping... is a homomorphism. Note: both groups are under addition. Please see attached for full question.

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 Fractional Transformation

1. Find the linear transformation that maps the points z_1 = 2, z_2 = i, z_3 = -2 onto the points w_1 = 1, w_2 = i, w_3 = -1. Ans. w = (3z + 2i)/(iz + 6)

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).

Matrices, Reflection Matrices and Linear Transformation

From the cubics P3 to the fourth degree polynomials P4, what matrix represents multiplication by 2 + 3t? The columns of the 5 by 4 matrix A come from applying the transformation to each basis vector x1 = 1, x2 = t, x3 = t2, x4 = t3. Verify directly from c2 + s2 = 1 that the reflection matrices satisfy H2 = I. c = cosine s

Transforming Representations Equations

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 Defining Cash-Flow Constraints

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

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 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

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.

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)