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# Linear Algebra

### Mappings, Homomorphisms and Subgroups

Let @:G-->H be a homomorphism of G onto H, and let N be a normal subgroup of G. Show that @(N) is a normal subgroup of H. How do I prove that a mapping is a normal subgroup of a group? What I am missing here is some understanding of the terminology and some clear understanding of mappings, homomorphisms, subgroups and no

### Mathematics - Algebra - Matrices Problems

Find the Eigen Values ( or Characteristic roots ) and Eigen Vectors ( or Characteristic vectors ) of the following matrix and give the Matrix which transforms A to diagonal form. -2 1 1 A = -11 4 5 -1 1 0 To see the question in its correct form, please download the attached que

### Linear Operator Finding Basis and Dimension of Range and Kernel For complete description of the problems, please see the questions.

Question (1): Let T: R^3 into R^3 be a linear transformation defined by T(x, y, z) = (x + 2y - z, y + z, x + y - 2z ) Find a basis and the dimension of ( i ) Range of T ( ii) the Kernel of T Question (2) : If T:R^4 into R^3 is a linear transformation defined by T( a, b, c, d) = ( a - b + c + d, a + 2c - d,

### Numerical Analysis : Solving Systems of Linear Equations

Problems 4.1 1, 2, 3, 6, 7, 8, 9, 11, 15 and 16. (page 147) I attached few pages description. See attached file for full problem description.

### Finding Basis and Dimension of the Subspace generated by given vectors. For complete description of the specific problems please see the problems.

Question (1) Find a basis and dimension of the subspace W of R4 generated by the vectors ( 1 , - 4 , - 2 , 1 ) , ( 1 , - 3 , - 1 , 2 ) , ( 3 , - 8 , - 2 , 7 ) . Extend it to find the basis of R4 Question (2) Determine a basis and the dimensions of the Subspace of M2(R) generated by the 2 by 2 Matrices [ 2 -10 ] , [

### Finding Basis and Dimension of the Subspace generated by given vectors. For complete description of the specific problems please see the problems.

Question (1) Find a basis and dimension of the subspace W of R4 generated by the vectors ( 1 , - 4 , - 2 , 1 ) , ( 1 , - 3 , - 1 , 2 ) , ( 3 , - 8 , - 2 , 7 ) . Extend it to find the basis of R4 . Question (2) Determine a basis and the dimensions of the Subspace of M2(R) generated by the 2 by 2 Matrices [ 2 -10 ] , [

### Non-Identity Elements of Prime Order

Suppose G is a finite group with the property that every non-identity element has prime order (D3 and D5 are examples of groups with this property). Show that if the center of G, Z(G), is not trivial, then every nonidentity element of G has the same order.

### Reduced Row Echelon Form of Homogeneous Systems of Equations

5. In general, a matrix's row echelon form can vary a bit. A matrix's reduced row echelon form is always unique. In other words, there is only one specific reduced row echelon form matrix associated with each matrix. (a) Consider the following homogeneous system: 2x1 ¡ x2 + x4 + 4x5 = 0 2x1 ¡ 2x2 + x3 + 4x4 ¡ 3x5 = 0 2x

### Math Questions: system of linear equations and inequalities

8.1 Exercises Solve each of the following systems by graphing. 10. 2x - y = 4 2x - y = 6 12. x - 2y = 8 3x - 2y = 12 20. 3x - 6y = 9 X - 2y = 3 26. Find values for m and b in the following system so that the solution to the system is (-3, 4). 5x + 7y = b Mx + y = 22 8.2 Exerc

### Trigonometry Word Problems and Expressing Linear Systems of Equations as Matrices

Researchers at the National Interagency Fire Center in Boise, Idaho coordinate many of the firefighting efforts necessary to battle wildfires in the western United States. In an effort to dispatch firefighters for containment, scientists and meteorologists attempt to forecast the direction of the fires. Some of this data can be

### Algebra : Matrices and Systems of Linear Equations (26 Problems)

Please see the attached files for the fully formatted problems. 1. Given the equation below, find f(x) where y = f(x). 8y(6x - 7) - 12x(4y + 3) + 265 - 5(3x - y + 2) = 0. 2. Solve these linear equations for x, y, and z. 3x + 5y - 2z = 20; 4x - 10y -z = -25; x + y -z = 5 3. The value of y in Question 2 lies in the ran

### Basic Algebra : Area, Solving Systems of Equations and Inequalities (40 Problems)

Please see the attached file for the fully formatted problems. 1. Write as a decimal. 2. Multiply. (-1)(5) 3. Give the coordinates of the point graphed below. 4. Solve. 8 - (4x - 3) = 19 5. A backyard has dimensions yards by yards. What is the area of the back yard in square yards (yd2)? 6. If a 4.4-pound bag of

### Exercise in logical thinking

There are 3 suspects, A, B, and C, for a robbery that presumably happened in a shop. We know that the following facts are true: (1) Each of A, B, C was in the shop on the day of the robbery, and no one else was there on that day. (2) If A was guilty, then he had exactly one accomplice. (3) If B is innocent, then so is

### Matrices : Row Echelon Forms and Systems of Linear Eqations

Row reduce matrix to reduced echelon from. Circle pivot positions in the final matrix and in the original matrix, and list the pivot columns. 1) Find the general solutions of the systems whose augmented matrices are given. 2) 3) Use the echelon form. Suppose each matrix represents the augmented mat

### Matrices and Linear Systems of Equations

The augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate operations and describe the solution set of the original system. 1) 2) Solve the system. 3) 4) Determine if the system is consistent. Don not completely solve b

### Basic Algebra Problems

See attached file for full problem description. 1. Graph the inequality. 2x + 3y > 6 2. Given g(x) = -3x + 5, find g(2a) 3. Graph the inequality: x - y < = 2 4. Graph the inequality: y > =3x 5. Given f(x) = -x^3 - 3x^2 -3x +9, find f(-2), f(0), and f(3) 6. Given f(x) = -5x - 1, find f(-2) 7. Graph f(x) = 4x + 1 8. Grap

### Question About Systems of Equations and Inequalities Word Problems

1. A coffee merchant has coffee beans that sell for \$9 per pound and \$12 per pound. The two types are to be mixed to create 100 lb of a mixture that will sell for \$11.25 per pound. How much of each type of bean should be used in the mixture? 2. In a town election, the winning candidate had 220 more votes than the lose

### system of linear equations and inequalities

Solve each system by the substitution method. Section 7.1 #54 X + 3y = 2 -x + y = 1 Section 7.2 #64 Book and magazines. At Gwen's garage sale, all books were one price, and all magazines another price. Harriet bought 4 books and three magazines for \$1.45 and June bought two books and five magazines for \$1.25. What was th

### Eigenvalue solutions of differential equations with boundary conditions.

Consider the equation: y'' + k*y = 0 with BC: y(0) = 0 , y(&#61552;) = a &#61625;0 Answer the following: 1. What are the restrictions on k such that there is a nontrivial solution? 2. Find a solution using eigenfunction expansion on [0,&#61552;] 3. Find a solution to the differential equation using any method 4. Co

### Using Linear Equations to Solve Problems

1. Sam needs to get carpet for two rooms of his house. Estimate the number of square feet (square footage) in the two rooms if one room measures 10 ½ feet by 9 ¾ feet; and the other room measures 19¼ feet by 18 ½ feet. 2. Sandra drove for 234.8 miles and used 12.6 gallons of gas. Estimate the number of miles Sandra's

### Eigenvalues : Stability of fixed points of linear differential systems.

Analyze the stability of the fixed points of linear differential systems. Provide several explicit examples, with graphical illustrations

### Solving Systems of Linear Equations and Matrices

1. Consider the system of equations x + y + 2z = a x + z = b 2x + y + 3z = c Show that for this system to be consistent, the constants a, b, and c must satisfy c = a + b. 2. Show that the elementary row operations do not affect the solution set of a linear system. 3. Consider the system of equations ax + by =

### Solving Linear Equations and Diameter of a Circle

Solve each linear equation. Show your work and check. 3(x-0.87)-2x=4.98 Solve each equation by first eliminating the decimal numbers. 0.08t+28.3=0.5t-9.5 Diameter of a Circle. If the circumference of a circle is 4pi meters, then what is the diameter?

### Linear Equations, Inequalities and Word Problems

Solve each equation and check your answer 2(a-4)+4=5(9-a) Solve each equation. Identify each equation as a conditional equation, an inconsistent equation or an identity. Solve each equation for y. Write the equation in the form of y=mx+b where m and b are real numbers Find the value of y in each formula if x=

### Expansion of a Function on an Interval with respect to Eigenfunctions

Find the expansion of f(x)=1-x on the interval [0,L], with respect to the eigenfunctions of the Sturm-Liouville problem: y''+&#955;y=0 y(0)=y(L)=0

### Linear Equations and Inequalities Word Problem

The basal energy requirement B is the number of calories that a person needs to maintain the life process. For a 28-year old female with a height of 160 centimeters and a weight of 45 kilograms (kg), B is 1300 calories. If her weight increases to 50 kg, then B is 1365 calories. There is a linear equation that expresses B in term

### Automorphisms

Show that all automorphisms of a group G form a group under function composition. Then show that the inner automorphisms of G, defined by f : G--->G so that f(x) = (a^(-1))(x)(a), form a normal subgroup of the group of all automorphisms. For the first part, I can see that we need to show that f(g(x)) = g(f(x)) for x in

### Solving Systems of Linear Equations

1) Solve by the addition method. 3x + 2y = 14 3x - 2y = 10 2) Solve by the addition method 5x = 6y + 50 2y = 8 - 3x 3) Solve. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 4) Can't type fractions, so

### Maximizing Area and Solving Systems of Linear Equations

1) You have 80 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area? 2) A rain gutter is made from sheets of aluminum that are 12 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that wil

### Radical Expressions and Linear Equations and Inequalities

14. 5x(x + 1)(x - 1) > 0 24. t(x) = - t(4), t(4), t(0), t(-1), and t(- ) 28. Find the domain of the function t in exercise 24. Graph. 42. f(x) = Simplify. 56. - For the given function, find the indicated function values. 60. g(x) = - g(-62), g(0), g(-13), an