Math

AlgebraElements Of Modern Algebraa. Prove part d of Theorem 1.30. b. Prove part e of Theorem 1.30. Theorem 1.30 □ Properties of Matrix Addition . Addition in M m × n ( R ) has the following properties. d . Each element of M m × n ( R ) has an additive inverse in M m × n ( R ) . e . Addition is commutative in M m × n ( R ) .BuyFind*arrow_forward*

8th Edition

Gilbert + 2 others

Publisher: Cengage Learning,

ISBN: 9781285463230

Chapter 1.6, Problem 16E

Textbook Problem

**a.** Prove part **d** of Theorem 1.30.

**b.** Prove part **e** of Theorem 1.30.

**Theorem 1.30**
**Properties of Matrix Addition**.

Addition in

**d**. Each element of

**e**. Addition is commutative in

Elements Of Modern Algebra

Show all chapter solutions

Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False
Label each of the following...Ch. 1.1 - True or False
Label each of the following...Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False
Label each of the following...Ch. 1.1 - True or False
Label each of the following...Ch. 1.1 - True or False Label each of the following...Ch. 1.1 - True or False Label each of the following...

Ch. 1.1 - For each set A, describe A by indicating a...Ch. 1.1 - 2. Decide whether or not each statement is true...Ch. 1.1 - Decide whether or not each statement is true. (a)...Ch. 1.1 - 4. Decide whether or not each of the following is...Ch. 1.1 - Evaluate each of the following sets, where U={...Ch. 1.1 - 6. Determine whether each of the following is...Ch. 1.1 - Write out the power set, (A), for each set A. a....Ch. 1.1 - 8. Describe two partitions of each of the...Ch. 1.1 - Write out all the different partitions of the...Ch. 1.1 - 10. Suppose the set has a .
a. How many elements...Ch. 1.1 - 11. State the most general conditions on the...Ch. 1.1 - 12. Let Z denote the set of all integers, and...Ch. 1.1 - 13. Let Z denote the set of all integers, and...Ch. 1.1 - In Exercises 1435, prove each statement. ABABCh. 1.1 - In Exercises 1435, prove each statement. (A)=ACh. 1.1 - In Exercises , prove each statement.
16. If and ,...Ch. 1.1 - In Exercises , prove each statement.
17. if and...Ch. 1.1 - In Exercises , prove each statement.
18.
Ch. 1.1 - In Exercises , prove each statement.
19.
Ch. 1.1 - In Exercises 1435, prove each statement. (AB)=ABCh. 1.1 - In Exercise 14-35, prove each statement.
21.
Ch. 1.1 - In Exercise 14-35, prove each statement. A(AB)=ABCh. 1.1 - In Exercises 14-35, prove each statement.
23.
Ch. 1.1 - In Exercise 14-35, prove each statement....Ch. 1.1 - In Exercise 14-35, prove each statement. If AB,...Ch. 1.1 - In Exercise 14-35, prove each statement.
26. If...Ch. 1.1 - In Exercise 14-35, prove each statement.
27.
Ch. 1.1 - In Exercise 14-35, prove each statement. A(BA)=Ch. 1.1 - In Exercises 14-35, prove each statement.
29.
Ch. 1.1 - In Exercises 14-35, prove each statement....Ch. 1.1 - In Exercises 1435, prove each statement....Ch. 1.1 - In Exercises 1435, prove each statement....Ch. 1.1 - In Exercises , prove each statement.
33.
Ch. 1.1 - In Exercises , prove each statement.
34. if and...Ch. 1.1 - In Exercises 1435, prove each statement. AB if and...Ch. 1.1 - Prove or disprove that AB=AC implies B=C.Ch. 1.1 - Prove or disprove that AB=AC implies B=C.Ch. 1.1 - 38. Prove or disprove that .
Ch. 1.1 - Prove or disprove that (AB)=(A)(B).Ch. 1.1 - 40. Prove or disprove that .
Ch. 1.1 - Express (AB)(AB) in terms of unions and...Ch. 1.1 - 42. Let the operation of addition be defined on...Ch. 1.1 - 43. Let the operation of addition be as defined in...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - True or False
Label each of the following...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - Label each of the following statements as either...Ch. 1.2 - 1. For the given sets, form the Cartesian...Ch. 1.2 - For each of the following mapping, state the...Ch. 1.2 - 3. For each of the following mappings, write out ...Ch. 1.2 - For each of the following mappings f:ZZ, determine...Ch. 1.2 - 5. For each of the following mappings, determine...Ch. 1.2 - 6. For the given subsets and of Z, let and...Ch. 1.2 - 7. For the given subsets and of Z, let and...Ch. 1.2 - 8. For the given subsets and of Z, let and...Ch. 1.2 - For the given subsets A and B of Z, let f(x)=2x...Ch. 1.2 - For each of the following parts, give an example...Ch. 1.2 - For the given f:ZZ, decide whether f is onto and...Ch. 1.2 - 12. Let and . For the given , decide whether is...Ch. 1.2 - 13. For the given decide whether is onto and...Ch. 1.2 - 14. Let be given by
a. Prove or disprove that ...Ch. 1.2 - 15. a. Show that the mapping given in Example 2...Ch. 1.2 - 16. Let be given by
a. For , find and .
b. ...Ch. 1.2 - 17. Let be given by
a. For find and.
b. For...Ch. 1.2 - 18. Let and be defined as follows. In each case,...Ch. 1.2 - Let f and g be defined in the various parts of...Ch. 1.2 - In Exercises 20-22, Suppose and are positive...Ch. 1.2 - In Exercises 20-22, Suppose and are positive...Ch. 1.2 - In Exercises 20-22, Suppose and are positive...Ch. 1.2 - Let a and b be constant integers with a0, and let...Ch. 1.2 - 24. Let, where and are nonempty.
Prove that for...Ch. 1.2 - 25. Let, where and are non empty, and let and ...Ch. 1.2 - 26. Let and. Prove that for any subset of T of...Ch. 1.2 - 27. Let , where and are nonempty. Prove that ...Ch. 1.2 - 28. Let where and are nonempty. Prove that ...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - True or False
Label each of the following...Ch. 1.3 - Label each of the following statements as either...Ch. 1.3 - For each of the following pairs and decide...Ch. 1.3 - For each pair given in Exercise 1, decide whether ...Ch. 1.3 - Let . Find mappings and such that.
Ch. 1.3 - Give an example of mappings and such that one of...Ch. 1.3 - Give an example of mapping and different from...Ch. 1.3 - 6. a. Give an example of mappings and , different...Ch. 1.3 - 7. a. Give an example of mappings and , where is...Ch. 1.3 - Suppose f,g and h are all mappings of a set A into...Ch. 1.3 - Find mappings f,g and h of a set A into itself...Ch. 1.3 - Let g:AB and f:BC. Prove that f is onto if fg is...Ch. 1.3 - 11. Let and . Prove that is one-to-one if is...Ch. 1.3 - Let f:AB and g:BA. Prove that f is one-to-one and...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - Label each of the following statements as either...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False Label each of the following...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False
Label each of the following...Ch. 1.4 - True or False Label each of the following...Ch. 1.4 - 1. Decide whether the given set is closed with...Ch. 1.4 - In each part following, a rule that determines a...Ch. 1.4 - 3. Let be a set of three elements given by . In...Ch. 1.4 - 4. Let be a set of three elements given by . In...Ch. 1.4 - 5. Let be the set of four elements given by with...Ch. 1.4 - Let S be the set of four elements given by S={...Ch. 1.4 - 7. Prove or disprove that the set of nonzero...Ch. 1.4 - 8. Prove or disprove that the set of all odd...Ch. 1.4 - 9. The definition of an even integer was stated in...Ch. 1.4 - 10. Prove or disprove that the set of all nonzero...Ch. 1.4 - Prove or disprove that the set B={ z3|zZ } is...Ch. 1.4 - 12. Prove or disprove that the set of non zero...Ch. 1.4 - Assume that is an associative binary operation on...Ch. 1.4 - Assume that is a binary operation on a non empty...Ch. 1.4 - 15. Let be a binary operation on the non empty...Ch. 1.4 - Assume that is an associative binary operation on...Ch. 1.5 - True or False Label each of the following...Ch. 1.5 - True or False Label each of the following...Ch. 1.5 - True or False
Label each of the following...Ch. 1.5 - For each of the following mappings exhibit a...Ch. 1.5 - 2. For each of the mappings given in Exercise 1,...Ch. 1.5 - 3. If is a positive integer and the set has ...Ch. 1.5 - 4. Let , where is nonempty. Prove that a has...Ch. 1.5 - Let f:AA, where A is nonempty. Prove that f a has...Ch. 1.5 - 6. Prove that if is a permutation on , then is a...Ch. 1.5 - Prove that if f is a permutation on A, then...Ch. 1.5 - 8. a. Prove that the set of all onto mappings from...Ch. 1.5 - Let f and g be permutations on A. Prove that...Ch. 1.5 - 10. Let and be mappings from to. Prove that if is...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - True or False
Label each of the following...Ch. 1.6 - Label each of the following statements as either...Ch. 1.6 - Write out the matrix that matches the given...Ch. 1.6 - 2. Perform the indicated operations, if...Ch. 1.6 - 3. Perform the following multiplications, if...Ch. 1.6 - Let A=[aij]23 where aij=i+j, and let B=[bij]34...Ch. 1.6 - 5. Show that the matrix equation is equivalent to...Ch. 1.6 - 6. Write a single matrix equation of the form ...Ch. 1.6 - Let ij denote the Kronecker delta: ij=1 if i=j,...Ch. 1.6 - Let S be the set of four matrices S={I,A,B,C},...Ch. 1.6 - 9. Find two square matrices and such that.
Ch. 1.6 - Find two nonzero matrices A and B such that AB=BA.Ch. 1.6 - 11. Find two nonzero matrices and such that.
Ch. 1.6 - 12. Positive integral powers of a square matrix...Ch. 1.6 - For the matrices in Exercise 12, evaluate (A+B)2...Ch. 1.6 - Assume that A1 exists and find a solution X to...Ch. 1.6 - 15. Assume that are in and with and invertible....Ch. 1.6 - a. Prove part d of Theorem 1.30. b. Prove part e...Ch. 1.6 - a. Prove part a. of Theorem 1.34. b. Prove part b....Ch. 1.6 - Prove part b of Theorem 1.35.
Theorem 1.35 ...Ch. 1.6 - Let a and b be real numbers and A and B elements...Ch. 1.6 - Prove that if then.
Ch. 1.6 - Suppose that A is an invertible matrix over and O...Ch. 1.6 - Let be the set of all elements of that have one...Ch. 1.6 - Prove that the set S={[abba]|a,b} is closed with...Ch. 1.6 - Prove or disprove that the set of diagonal...Ch. 1.6 - Let A and B be square matrices of order n over...Ch. 1.6 - Let and be square matrices of order over ....Ch. 1.6 - A square matrix A=[aij]n with aij=0 for all ij is...Ch. 1.6 - Let a,b,c,andd be real numbers. If adbc0, show...Ch. 1.6 - Let A=[abcd] over . Prove that if adbc=0, then A...Ch. 1.6 - Let be elements of where is not a zero matrix....Ch. 1.6 - Let A,BandC be square matrices of order n over ....Ch. 1.6 - Let A and B be nn matrices over such that A1 and...Ch. 1.7 - Label each of the following statements as either...Ch. 1.7 - True or False
Label each of the following...Ch. 1.7 -
True or False
Label each of the following...Ch. 1.7 - Label each of the following statements as either...Ch. 1.7 - True or False
Label each of the following...Ch. 1.7 - Label each of the following statements as either...Ch. 1.7 - For determine which of the following relations...Ch. 1.7 - 2. In each of the following parts, a relation is...Ch. 1.7 - a. Let R be the equivalence relation defined on Z...Ch. 1.7 - 4. Let be the relation “congruence modulo 5”...Ch. 1.7 - 5. Let be the relation “congruence modulo ”...Ch. 1.7 - In Exercises 610, a relation R is defined on the...Ch. 1.7 - In Exercises 610, a relation R is defined on the...Ch. 1.7 - In Exercises 610, a relation R is defined on the...Ch. 1.7 - In Exercises 610, a relation R is defined on the...Ch. 1.7 - In Exercises , a relation is defined on the set ...Ch. 1.7 - Let be a relation defined on the set of all...Ch. 1.7 - Let and be lines in a plane. Decide in each case...Ch. 1.7 - 13. Consider the set of all nonempty subsets of ....Ch. 1.7 - In each of the following parts, a relation is...Ch. 1.7 - Let A=R0, the set of all nonzero real numbers, and...Ch. 1.7 - 16. Let and define on by if and only if ....Ch. 1.7 - In each of the following parts, a relation R is...Ch. 1.7 - Let (A) be the power set of the nonempty set A,...Ch. 1.7 - For each of the following relations R defined on...Ch. 1.7 - Give an example of a relation R on a nonempty set...Ch. 1.7 - 21. A relation on a nonempty set is called...Ch. 1.7 - A relation R on a nonempty set A is called...Ch. 1.7 - A relation R on a nonempty set A is called...Ch. 1.7 - For any relation on the nonempty set, the inverse...Ch. 1.7 - 25. Let , , and . Write out and .
Ch. 1.7 - 26. Let , , and . Write out and .
Ch. 1.7 - Prove Theorem 1.40: If is an equivalence relation...Ch. 1.7 - 28. Let , and . Define the relation R on A by if...Ch. 1.7 - 29. Suppose , , represents a partition of the...Ch. 1.7 - Suppose thatis an onto mapping from to. Prove that...

Find more solutions based on key concepts

Show solutions For problems 27-54, simplify each of the numerical expressions. Objective 2 and 3 4(1)23(2)3

Intermediate Algebra

The tangent line to a curve at a point closely approximates the curve near the point. In fact, for x-values cl...

Mathematical Applications for the Management, Life, and Social Sciences

For a population with a mean of =40 and a standard deviation of =8 , find the z-score corresponding to each of ...

Statistics for The Behavioral Sciences (MindTap Course List)

To locate the centroid of a triangle, is it necessary to construct all three medians?

Elementary Geometry For College Students, 7e

For the following set of stores, find the value of each expression: a. X b. (X) c. (X 3) d. (X 3) X 4 5 2 1 3

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Suppose identical tags are placed on both the left ear and the right ear of a fox. The fox is then let loose fo...

Probability and Statistics for Engineering and the Sciences

Convert the expressions in Exercises 8596 radical form. 22/3

Finite Mathematics and Applied Calculus (MindTap Course List)

Use the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain w...

Single Variable Calculus

Convert each expression in Exercises 25-50 into its technology formula equivalent as in the table in the text. ...

Finite Mathematics

You have been observing the local Democratic Party in a large city and have compiled some information about a s...

Essentials Of Statistics

In Exercises 516, evaluate the given quantity. log22

Applied Calculus

Evaluate the double integral. 18. D(x2+2y)dA, D is bounded by y = x, y = x3, x 0

Calculus: Early Transcendentals

Newtons Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass...

Single Variable Calculus: Early Transcendentals, Volume I

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates. 52. r = 3

Precalculus: Mathematics for Calculus (Standalone Book)

In Exercises 15 to 18, show that the given set is an infinite set by placing it in a one-to-one correspondence ...

Mathematical Excursions (MindTap Course List)

In Exercises 1-6, simplify the expression. 5. (4x1)(3)(3x+1)(4)(4x1)2

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Graphing Linear Equations In Exercises 45-52, find the slope and y-intercept (if possible) of the equation of t...

Calculus: An Applied Approach (MindTap Course List)

Use the graphs you made in Problems 13 through 18 to find all values of x,0x2, for which the following are true...

Trigonometry (MindTap Course List)

If the minute hand of a clock has length r in centimeters, find the rate at which it sweeps out area as a funct...

Calculus (MindTap Course List)

Change from rectangular to spherical coordinates. 4. (a) (1, 0, 3) (b) (3, 1, 23)

Multivariable Calculus

Sweets May Not Be Culprit in Hyper Kids was a USA Today (February 3. 1994) headline reporting results of a stud...

Understanding Basic Statistics

Finding Unit Vectors In Exercises 79-82, find a unit vector (a) in the direction of t and (b) in the direction ...

Calculus: Early Transcendental Functions

Find each quotient and check: 2x3+3x29x+5x+3

Elementary Technical Mathematics

Calculate the value of the nonforfeiture options for the following life insurance policies. Face Value Years in...

Contemporary Mathematics for Business & Consumers

Using the Trapezoidal Rule and Simpson's Rule In Exercises 110, use the Trapezoidal Rule and Simpsons Rule to a...

Calculus: Early Transcendental Functions (MindTap Course List)

In Exercises 5964, sketch the straight line defined by the linear equation by finding the x- and y- intercepts....

Finite Mathematics for the Managerial, Life, and Social Sciences

Sketch the curve and find the area that it encloses. 10. r = 1 sin

Single Variable Calculus: Early Transcendentals

Fluid Force Evaluate the following two integrals, which yield the fluid forces given in Example 6. (a) Finside=...

Calculus of a Single Variable

The report Digital Democracy Survey (Deloitte Development LLC, 2016, deloitte.com/us/en.html, retrieved Novembe...

Introduction To Statistics And Data Analysis

In the graph at the right, limx1f(x)=. a) 2 b) 1 c) 3 d) does not exist

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

By the method of undetermined coefficients, a particular solution to y″ − 5y′ + 6y = e4x will have the form:
yp...

Study Guide for Stewart's Multivariable Calculus, 8th

In Exercises 9 to 28, use the drawings as needed to answer the following questions. Using the number line provi...

Elementary Geometry for College Students

Fill in the blanks. To find the _________ of a rational function, let x=0 and solve for y or find f(0).

College Algebra (MindTap Course List)

Actuaries are analysts who specialize in the mathematics of risk. Actuaries often work for insurance companies ...

Statistics for Business & Economics, Revised (MindTap Course List)

In each of the following exercises, the value on the left must be multiplied by one of the following numbers: 0...

Mathematics For Machine Technology

Define counterbalancing and explain how it is used to minimize or eliminate threats to internal validity from t...

Research Methods for the Behavioral Sciences (MindTap Course List)

Ten major college football bowl games were played in January 2010, with the University of Alabama beating the U...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Using Basic Integration Rules In Exercises 18, use the basic integration rules to find or evaluate the integral...

Calculus (MindTap Course List)

Define fraud and explain the safeguards that exist to prevent it.

Research Methods for the Behavioral Sciences (MindTap Course List)

CONCEPT CHECK Method of Lagrange Multipliers In your own words, describe the Method of Lagrange Multipliers for...

Multivariable Calculus

Reminder Round all answers to two decimal places unless otherwise indicated. Poiseuilles law for fluid velociti...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

Under what conditions is a linear function f(x) = mx + b, m 0, a linear transform?

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

Convert 315708 to base two.

Mathematics: A Practical Odyssey

A single line divides a plane into two regions. Two lines (by crossing) can divide a plane into four regions, t...

Discrete Mathematics With Applications

12. A survey question for a sample of 150 individuals yielded 75 Yes responses, 55 No responses, and 20 No Opin...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

Ambulance Calls by Day of Week. An ambulance service responds to emergency calls for two counties in Virginia. ...

Essentials Of Statistics For Business & Economics

Sketching a Transformation of a Rational Function In Exercises 7-10, use the graph of f(x)=3x2 to sketch the gr...

College Algebra