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# Linear Algebra

### Lebesgue Measure

6.T. Show that if f∈L∞(X,X,µ), then |f(x)|≤∥f∥∞ for almost all x. Moreover, if A <∥f∥∞then there exists a set E∈X with µ(E)>0 such that |f(x)|>A for all x∈E. 6.U. Show that if f∈Lp, 1≤p≤∞, and g∈L∞, then the product fg&#

### Linear Programming, Confidence Levels, Probability and Systems of Equations

29 Problems Finite Math. See attached file for full problem description.

### Finite Mathematics

Please help with these problems. See attached file for full problem description. 1. Find 2. a) Find the inverse of: . b) Use it to solve the following system of equations: 2x1 + x2 + x3 = 4 x2 + 2x3 = 0 2x1 + 2x2 + x3 = 1.

### How the row operations correspond to operations on the underlying system of equations.

The three elementary row operations come from a very natural place - they are the matrix equivalent of the same three operations allowed when reducing and solving a system of equations. See attached file for full problem description. Clearly explain, matrix by matrix, how the row operations correspond to operations on t

### 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

### Systems of Congruences

Find the necessary conditions on r1,r2 and n1,n2 that will guarantee a solution to the system: x ≡ r1 mod n1 x ≡ r2 mod n2. keywords: congruent, cogruency

### Diophantine Equations

Prove that y^2= x^3+23 has NO integer solutions.

### 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

### Use the given facts to determine whether or not a specified event occurred.

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

### Solving Systems of Equations

Find values for M and B in the following system so that the solution is (-3,4) 5X + 7Y=B MX + Y=22

### Zorn's Lemma

Prove the real numbers, R, contain a subring A with 1 Є A and A maximal (under inclusion) with respect to the property that 1/2 not Є A [Use Zorn's theorem]. See attached file for full problem description.

### Systems of Equations and Inequalities and Absolute Values

#14 Solve each system by the substitution method. Indicate whether each system is independent, inconsistent, or dependent. #24 Graph and solve the absolute value equality. #46 Graph each compound or absolute value inequality. |x-3y| ≥ 9 Please see the attached file for the fully formatted problems.

### 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

### Ill-Conditioned Systems of Equations

The solution of following system of linear equation is: 2x-2y+3z=1 5x-6y+10z=2 2x-4y+9z=1 x=2, y=3, z=1 And the solution of this system is: 2x-2y+3z=1 5x-6y+10.1z=2 2x-4y+9z=1 x=2.375 y= 3.75, z= 1.25 (You do not have to show this) Comparing the 2 solution can you determine whether the fist system of equat

### 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

### Irreducible Polynomials : Prove that there exist at most 2 non-isomorphic fields of order 4.

Prove that there exist at most 2 non-isomorphic fields of order 4.

### 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 =

### 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=

### Prove the identity

Please explain how to prove the identity with the following example: sin x - sin 3x / cos x - cos 3x = -cot 2x

### Congruences and Primes

Prove that is p is prime, we have: n choose m is congruent to [floor(n/p) choose floor(m/p)]*[(n mod p) choose (m mod p)] (mod p) Hint: show that (1+x)^(pq+r) is congruent to (1+x)^r * (1+x^p)^q (mod p) If you can point me to a book or website explaining how to do this type of problem, and give a sketch of the proof,

### 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