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Linear Algebra

Conjugacy Classes

Let K={k1,....km} be a conjugacy class in the finite group G. a) Prove that the element K=k1+k2+....km is the center of the group ring R[G] (check that g^-1Kg=K for all gin G) b) Let K1,....Kr be the conjugacy classes of G and for each Ki let Ki be the element of R[G] that is the sum of the members of Ki. Prove that an el

Applications of Linear Equations

1. An art dealer sold two artworks for $1520 thereby making a profit of 25% on the first work and 10% profit on the other, whereas if he had approached any exhibition he would have sold them together for $1535 with a profit of 10% on the first and 25% on the other artwork. Find the actual cost of each artwork. 2. A professor

Systems of Equations

Find a linear function f(x) = mx + b whose graph has the given slope and y-intercept. 6. Slope: 4/5 ; y-intercept: (0,28) Find an equation of the line having the given slope and containing the given point 14. m = 3, (-2, -2) Find an equation of the line containing the given pair of points. 28. (-4, -7) and (-2, -1

Solving System of Linear Equations Characteristics

1. Find k if the following system of equations has a unique solution 2x + (k - 1)y = 6 3x + (2k + 1)y = 9 2. Find k if the following system of equations has infinite solutions kx + 3y = k - 3 12x + ky = k 3. Find k if the following system of equations has no solution 3x + 2y = 6 kx + (k - 1)y = 9

Solve for F

See attached file for full problem description. When converting from Fahrenheit degrees to Celsius degrees, a well known formula is used: C= 5(F - 32) / 9

Coordinate system

See attached file for full problem description. 1. Find the slope of the line passing through the points (-8, -3) and (-2, 2). A) B) C) D) 2. Give the coordinates of the point graphed below. A) (4, 0) B) (-4, 0) C) (0, 4) D) (0, -4) 3. Find the slope of the graphed line.

Why every odd order subgroup has to be subgroup of cyclic odd

Please see the attached file first. I did the proof, but it's weak, since I can't find a way to argue why every odd order subgroup has to be a subgroup of the cyclic odd order subgroup K of index 2, and by the divisibility argument it still could be a subgroup of G and not to be contained entirely in K.

Solving Systems of Linear Equations Word Problems

Jill, Karen, and Betsy studied a total of 93 hours last week. Jill's and Karen's study time totaled only one-half as much as Betsy's. If Jill studied 3 hours more than Karen, then how many hours did each one of the girls spend studying?

Matrices, Eigenvectors, Eigenvalues and Inverses

Determining the equation for a matrix and confirming the inverse, eigenvalues Details: I have think the answer to question (a) is theta^2 [(1-p)I + pJ]. But when I am multiplying the sigma and the sigma inverse I still have "stuff" at the tail end of the identity......being added. I know I should be left with the identity if t

Prove that every regular tournament is strong.

Prove that every regular tournament is strong. Hints Here we need to first figure out something more about outdegrees and indegrees and orders of regular digraphs. Try to find a regular digraps with 3, 4, 5 ,6 vertices, and generalize. D is a regular tournament if there is k such that outdegree x = k and indegree x = n-k

Solving Systems of Linear Equations Word Problems

Flying against the jetstream, a jet travels 1890km in 3 hours. Flying with the jetstream, the same jet travels 4650km in 5 hours. What is the speed of the jet in still air, and what is the speed of the jetstream?

Systems of Equations Word Problems

Two rainstorms occurred in one week in a certain area. In the first rainstorm 35ml of rain fell per hour, and in the second rainstorm 20ml of rain fell per hour. Rain fell that week for a total of 45 hours for a total rainfall of 1200ml. How long was each of the two rainstorms?

Forest and subgraph

3.2 prove that a graph G is a forest if and only if every induced subgraph of G contains a vertex of degree at most 1. Please can you explain in here when the graph G is a forest and induced subgraph.

Systems of equatinos

1. (Solve system by the substitution method. Determine whether the equations are independent, dependent or inconsistent.) 2x - y = 3 2y = 4x - 6 2. (Solve system by the substitution method. Determine whether the equations are independent, dependent or inconsistent.) y = 3(x - 4) 3x - y = 12 3. (Solve sy

Solving Systems of Linear Equations: Example Problem

Solve each system by the addition method. Indicate whether each system is independent, inconsistent, or dependent. 1. -3x + y = 3 2x - 3y = 5 2. x + 3(y - 1) = 11 2(x - y) + 8y = 28 3. 1/3x - 1/6y = 1/3 1/6x + 1/4y = 0

Solving Word Problems using Systems of Equations

Flying against the jetstream, a jet travels 2600km in 4 hours. Flying with the jetstream, the same jet travels 6060km in 6 hours. What is the speed of the jet in still air, and what is the speed of the jetstream?

Systems of Equations Word Problems

A child in an airport is able to cover 380 meters in 4 minutes running at a steady speed down a moving sidewalk in the direction of the sidewalk's motion. Running at the same speed in the direction opposite to the sidewalk's movement, the child is able to cover325 meters in 5 minutes. What is the child's running speed on a sti

Systems of Equations Word Problems

In a family there are two cars. In a given week, the first car gets an average of 40 miles per gallon, and the second car gets 25 miles per gallon. The two cars combined drive a total of 1775 miles in that week, for a total gas consumption of 50 gallons. How many gallons were consumed by each of the two cars that week?

Systems of Equations Word Problems: Tickets Sold

A total of 200 tickets were sold for the carnival to adults and children. Children tickets were $6 and adult tickets were $8. If the total revenue for the carnival is $550 dollars, then how many tickets of each type were sold?

Find the linear equation.

56. If you earned an average of $25,000 over your working life and you retire after 2005 at age 62, 63, or 64, then your annual Social Security benefit will be $7,000, $7500, or $8000 respectively (www.ssa.gov). There is a linear equation that gives the annual benefit b in terms of age a for these three years. Find the equation.

Systems of Linear Inequalities

1) What conditions need to be satisfied for a solution of a system of linear inequalities to have it's solution in the first quadrant? Create an example of such a system. 2) Can you give a real world example when the solution of a system of inequalities must be in the first quadrant? 3) Suppose, wheat and sugar are two b

Show that every tree is central or bicentral.

1.- A tree is central if its center is K1 and bicentral if its center is K2. Show that every tree is central or bicentral. ( Kn is call complete graph and it if every two of its vertices are adjacent and every vertex has degree n) 2.- Prove taht a tree with Delta(T)=k ( Delta means maximum degree) has at least k vertices of d

Probs in Linear Algebra

1 3 11. Consider (A) = 2 1 (a) Find the eigenvalue and corresponding eigenvectors of A. (b) Determine matrices B and C such that B A C is diagonal. (c) Show the eigenvectors of (A) are linearly independent. (d) Repre

Solve each system by the substitution method.

Solve each system by the substitution method. Indicate whether each system is INDEPENDENT, INCONSISTENT or DEPENDENT 1. x+3(y-1)= 11 2(x-y)+8y+28 2. 1/3x-1/6y+1/3 1/6x+1/4y=0 Solve each compound inequity. State the solution set using interval notation and graph it. 3. x <=0 and x+6>3 and 1/4x>3 4. 1/3x>

linear equations and inequality

Complete each problem show work and ad graphs as needed John spent 1/3 of his inheritance on loose women, 1/5 on expensive cigars, and 3/10 on liquor. He invested the rest in lottery tickets. What percent of his inheritance went to lottery tickets? Evaluate the expression: -(2 - 3)4 Fred and Ethel are florists. They a

Solving Linear Systems of Equations

1. INDEPENDENT LINEAR SYSTEM a. a system with exactly one solution b. an equation that is satisfied by every real number c. equations that are identical d. a system of lines 2. DEPENDENT SYSTEM a. a system that is independent b. a system that depends on a variable c. a system that has no soluti

Solving Systems of Linear Equations by the Echelon Method

4. solve by gauss jordan method. 28.6x + 94.5y + 16.0z - 2 .94w = 198.3 16.7x + 44.3y - 27.3z + 8.9w = 254.7 12.5x - 38.7y +92.5z +22.4w =562.7 40.1x - 28.3y + 17.5z - 10.2w = 375.4 5. Solve by echelon method 4x - y + 3z = -2 3x + 5y - Z =15 -2x+y + 4z = 14 6. Solve the system of equation...let z be the