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# Linear Algebra

### Determine the real general solution to the system dx/dt=Ax.

Let A be a 2x2 real matrix having an eigenvalue lamda = 3 + 2i and corresponding eigenvector (1, -1-i). Then the real general solution to dx/dt = Ax is: Please refer to the attachment for the choices. Please show detailed working as to how you arrived at the answer.

### Proof: Generalization of the last Sylow's theorem

Prove the following generalization of the last Sylow's theorem: If |G| is divisible by p^b, and H <= G has order p^a with a <= b, then the number of subgroups of G that both contain H and have order p^b is congruent to 1 modulo p?

### Finding maximal ideal

A) Show that there is exactly one maximal ideal in Z_8 and in Z_9. b) Show that Z_10 and Z_15 have more than one maximal ideal.

### Formulation of System of Equations

Please see the attachment.

### System of Equations

We described three methods to solving Linear Systems with two Equations. They are the Graph method, the Elimination method, and the Substitution method. When would you use each method? What makes each method better than the other methods?

### Normal Sylow Theorem

Let G be a group of order 48. By the 1st Sylow theorem G has a Sylow 2-subgroup and a Sylow 3-subgroup. Suppose none of these are normal. Determine the number of Sylow 2-subgroups and Sylow 3-subgroups that G can have. Justify your answer.

### Sylow 5-subgroup ordering

Suppose that G is a group of order 30 and has a Sylow 5-subgroup that is not normal. Find the number (not what they are) of elements of order 1, of order 2, of order 3, of order 5. Justify your answer.

### Wieferich m-squares

It is a fact that for any odd prime p, there is a number b so that b has order p(p-1) modulo p^2. Given that fact, show that for any odd prime p, there is a number m so that p^2 is a Wieferich m-square.

### Linear Equation in Two Variables: Line 1, 2 or Neither

For each point in the table below, decide whether it is on Line 1, Line 2, both, or neither. Line 1: -10x + 13y = 12 Line 2: -10x + 7y = -12 (4,4) (-8, 4) (-3, -6).

### Show thw equivalence of two definitions of a contractible space.

We have defined space X to be contractible in two ways: Definition 1: X is contractible if it is homotopy equivalent to a point; and Definition 2: X is contractible if the identity map of X is null-homotopic. Show that these two definitions are equivalent.

### Algebra Problems for Circular Cylinder Hemispheres

Please solve as follows: • o For noninteger answers, please write your answer as a fraction rather than a decimal. • To show your work, you will need to include o the algebra used to compute the solution to any equations. o the formula with substituted values. o the final calculated ans

### Fundamental groups of the Moebius strip and the cylinder.

Show that the Mobius strip and the cylinder both have fundamental group Z. We can use the following theorem: If G acts on X, pi1(X) = {e}, and for all x elements of X there exists Ux neighborhood of X such that Ux intersection g(Ux) = empty set for all g elements of G{e}, then pi1(XG) is homeomorphic to G.

### Linear algebra help

32. Answer the following: a. Let (where is the vector space of 2 by 2 matrices.) Find an example that shows U is NOT a subspace of . The matrix where U=[1 0; 0 1] the det=1. This would not be a subspace of M22. b. Let where is the vector space of real-valued functions defined on the interval [0, 2]. Show that

### graphing, substitution, elimination, matrix

Part I (refer to the section on Systems of Linear Equations; Matrices in your text): As a restaurant owner there are many decisions that you need to make on a daily basis, such as where to keep inventory levels. You wish to replenish your stock of dishes by purchasing 250 sets for your restaurant. You have two dish design

### Prove/Disprove properties of subgroups

Let A be a group and let Z be a subgroup of A. Let x,y be elements of G. Tell if each statement is true of false and give reason 1) if Zx = Zy then xZ = yZ 2) if Zx = Zy then Zxy^-1 = Z 3) if Zx = Zy then Zx^2=Zy^2

### Linear Programming (Primal Problem) - Simplex Method

Please see attached documents for further details. Consider the following linear program (primal problem): Minimize f(x1; x2) = 10x1 + 14x2 subject to x1 + 2x2 â?¥ 3 2x1 + x2 â?¥ 4 3x1 + x2 â?¥ 2 x1, x2 â?¥ 0. a. Set up the dual of the above linear program. b. Use the simplex method to solve the d

### Solving for Systems of Linear Equations

Create a system of linear equations from my own life. Keep in mind that a system of linear equations will consist of two equations using the same variables and the variables will represent the same thing for both equations, i.e., if you use X as a variable in the first equation and X represents a number of Beers then you need to

### Systems and Equations

1. What is a system of equations? 2. Solve for X and Y in the following problems using either substitution or elimination methods. Make sure you show all your work so you can get partial credit even if your final answer is wrong. a. X + Y = 10 , 3X + Y = 12 b. 2X + 5Y = 19 , 3X + 3Y = 15 c. 4X + Y = 22 ,

### Finding the Interior and the Closure of a Set

Please see attached file. Please help write a rigorous proof on the first problem, which is called the EASY problem. Consider the set A in the attached file. Compute, for the euclidean metric, the interior and the closure of A in R2. If you consider A as a subspace, is it complete?

### Linear Equations & Functions

Good morning - my son has chapter test tomorrow and on linear equations. I would love to be able to help him study, unfortunately is been a while since I've done linear equations and my knowledge is sketchy at best. Attached is a study guide that he will be using to study from. Can you please solve the equations and provi

### calculate QR

0 30 10 20 = P 20 20 10 2 0 2 20 50 1 20 2 = Q 180 = R 60 25 QR= 10

### Describe how to solve a 3x3 linear system. Provide example

I require assistance with the following short answer question. Response should be at least one paragraph with at least one logically sound example. > Describe how to solve a 3x3 linear system. Provide one example.

### Using Fermat's Little Theorem

Please DO NOT USE Euler's function. You can only use Fermat's Little theorem if needed. Problem: Prove that if p is prime and (a,p)=1 , then the congruence ax = b (mod p) has the solution x = a^(p-2) b (mod p) Thank you!

### Real eigenvalue of A, and v the associated eigen-vector.

Let A be an nxn matrix, λ a real eigenvalue, v the associated eigenvector. Show that A^k v= λ^k v. Use this to show e^At v= e^λt v. Use this to show that the line through the origin consists of two solutions to x'=Ax.

### Jordan Canonical form of a matrix

For the given 4x4 matrix, find P such that INV(P)AP is in Jordan Canonical Form. A = | 2 1 0 0 | | -1 4 0 0 | | 0 0 2 1 | | 0 0 -1 2 | It is easy to find repeated eigenvalues (3, 2+i, 2-i). If I treat the upper block as a 2x2 matrix, I can find P = [1 0; 1 1] (Note: Using Matlab notation

### Linear algebra

(A paradox) Suppose A has a right-inverse B. Then AB = I ... see the picture please. Thank you.

### Linear Trend Equations in Business Scenarios

1. The linear trend equation: Y' = a + bt, where b is the slope and a is the y-intercept, is an example of a linear function. A. Given that Y' = 1000, b = 10, what is the y-intercept when the independent variable time, t = 65? B. What might be predicted when t = 24 C. We have another linear trend equatio

### Ranges of grade for variation and quantity

1) a student must have an average (the mean) on five tests that is greater than or equal to 80% but less than 90% to receive a final grade of B. Devons grades on the first four tests were 76%,63%,91%, and 80%. What range of grades on the fifth test would give him a B in the course? (assume that 100% is the highest possible grade

### System of Linear Equations in Real Life

Create a system of linear equations from your own life and write a brief paper describing the system. Is this a consistent system or inconsistent system?

### Functions and Models

Think of a mathematical function that represents something from your own life. For example, suppose the number of beers you drink depends on the number of football games you watch. If you drink five beers during every football game, the function would be: Number of Beers (B) = 5 times Number of Football Games (F), or B = 5F