The Heart of Mathematics: An Invitation to Effective Thinking
4th Edition
ISBN: 9781118156599
Author: Edward B. Burger, Michael Starbird
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 8.2, Problem 20MS
High roller (H). Using two fair dice, what is the prbability of rolling a sum that exceeds 4?
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3. Define a function T : R³ → R² by
T(x, y, z) = (x + y + z, x + 2y − 3z).
(1) Show that T is a linear transformation.
—
(2) Find all vectors in the kernel of T.
(3) Show that T is onto.
(4) Find the matrix representation of T relative to the standard ba-
sis of R³ and R².
5. Use the Gram-Schmidt process to find an orthonormal basis for the
subspace of R4 spanned by the vectors
(1,0,1,1),
(1,0,1,0),
(0, 0, 1, 1).
4. Show that B
=
{(1, 1, 1), (1, 1, 0), (0, 1, 1)} is a basis for R³. Find
the coordinate vector of (1, 2, 3) relative to the basis B.
Chapter 8 Solutions
The Heart of Mathematics: An Invitation to Effective Thinking
Ch. 8.1 - Doors galore. The 21st-century version of Lets...Ch. 8.1 - Birthday surprise. How many people would you need...Ch. 8.1 - Opposite of heads. Suppose you flip a coin 100...Ch. 8.1 - Penny percent. Suppose you flip a penny 50 times,...Ch. 8.1 - Party time. At a nephews party, you decide to...Ch. 8.1 - Flipping Lincoln. Flip a penny 100 times and...Ch. 8.1 - Flashing cards. Shuffle a standard deck of 52...Ch. 8.1 - King for a day. Remove three cards from a deck of...Ch. 8.1 - A card deal stick. Remove three cards from a deck...Ch. 8.1 - A card deal switch. Remove three cards from a deck...
Ch. 8.1 - A card reunion—black first (S). Using a shuffled...Ch. 8.1 - A card reunion (H). Using a shuffled deck of...Ch. 8.1 - Prob. 14MSCh. 8.1 - Prob. 15MSCh. 8.1 - Personal perspectives. Write a short essay...Ch. 8.1 - With a group of folks. In a small group, discuss...Ch. 8.1 - One Grecian urn. A large urn in your kitchen is...Ch. 8.1 - Two Grecian urns. Your math instructor keeps two...Ch. 8.1 - Pennies from heaven (H). Residents of your...Ch. 8.1 - Dating Penny. The pennies described in the...Ch. 8.1 - Changing your pants. You have 20 coins in your...Ch. 8.2 - Black or white? Your friend chooses his sartorial...Ch. 8.2 - Eleven cents. You have a dime and a penny. Flip...Ch. 8.2 - Yummm. You have a small bag of candy-coated...Ch. 8.2 - Rubber duckies. A game at a carnival has 75 rubber...Ch. 8.2 - Legally large. What does the Law of Large Numbers...Ch. 8.2 - Lincoln takes a hit. On your wall is a poster...Ch. 8.2 - Giving orders. Order the following events in tenns...Ch. 8.2 - Two heads are better. Simultaneously flip a dime...Ch. 8.2 - Tacky probabilities. Before doing the following...Ch. 8.2 - BURGER AND STARBIRD. Suppose you randomly select a...Ch. 8.2 - Monty Hall. Read and rework the Lets Make a Deal...Ch. 8.2 - 7 or 11 (S). What is the probability of rollign a...Ch. 8.2 - D and D. You simultaneously flip a dime and roll a...Ch. 8.2 - The top 10 (ExH). Suppose you have 10 marbles....Ch. 8.2 - One five and dime (II). Someone simultaneously...Ch. 8.2 - Five flip. Someone flips five coins, but you dont...Ch. 8.2 - Flipped out. We take a coin and flip it 10,000,000...Ch. 8.2 - Spinning wheel. A roulette wheel has 36 spaces...Ch. 8.2 - December 9. Choose two people at random. What is...Ch. 8.2 - High roller (H). Using two fair dice, what is the...Ch. 8.2 - Double dice. You roll two fair dice. What is the...Ch. 8.2 - Sifly puzzle. After a professor explains the...Ch. 8.2 - Just do it. Find groups of roughly 35 people...Ch. 8.2 - No matches (S). Suppose 40 people are in a room....Ch. 8.2 - Spinner winner. If you were to spin the wheel...Ch. 8.2 - Flip side (S). Someone flips three coins behind a...Ch. 8.2 - Other flip side. Someone flips three coins behind...Ch. 8.2 - Blackjack. From a regular deck of 52 playing...Ch. 8.2 - Be rational (ExH). Suppose someone has randomly...Ch. 8.2 - Well red (H). Someone shows you three cards. One...Ch. 8.2 - Regular dice. Dungeons and Dragons players use...Ch. 8.2 - Take your seat. You decide to fly to California on...Ch. 8.2 - Eight flips. What is the probability of flipping a...Ch. 8.2 - Lottery (S). The lottery in an extremely small...Ch. 8.2 - Making the grade. What is wrong with the following...Ch. 8.2 - Dont squeeze. Five shoppers buy Charmin toilet...Ch. 8.2 - Birthday cards. Using a regular deck of 52 playing...Ch. 8.2 - Too many boys. Long, long ago and far, far away,...Ch. 8.2 - Three paradox (H). The correct probability of...Ch. 8.2 - Whats the big event? The section defines the...Ch. 8.2 - Tallying totals. For each given probability below,...Ch. 8.2 - Algebra roulette. Your math instructor has created...Ch. 8.2 - Probability beans. The jellybean jar on your math...Ch. 8.2 - Picking up the tab (H). Ed and Mike go out to...Ch. 8.3 - Daily deaths. About 58 million people die every...Ch. 8.3 - Wake-up call. Suppose you wake up each morning and...Ch. 8.3 - More than 12 monkeys. What does the Infinite...Ch. 8.3 - Get shorty. Who was Georges Louis Leclerc Comte de...Ch. 8.3 - Nothing but heads. If you flip a fair coin three...Ch. 8.3 - Pick a number. Pick a number from the following...Ch. 8.3 - Personal coincidences. List three coincidences you...Ch. 8.3 - No way. It is the last Sunday of spring break and...Ch. 8.3 - Prob. 10MSCh. 8.3 - Unlucky numbers. Suppose you randomly picked 1000...Ch. 8.3 - A bad block (S). Suppose 1054 people died in...Ch. 8.3 - Coin toss test. Ask two friends to help you repeat...Ch. 8.3 - DeceptIve dice (ExH). You asked three friends to...Ch. 8.3 - Murphys Law. If something can go wrong, it will....Ch. 8.3 - Drop the needle. Try Buffons needle experiment by...Ch. 8.3 - IBM again (H). Suppose we try the IBM...Ch. 8.3 - The dart index. Take a page of stock quotes from a...Ch. 8.3 - Random walks. Using a piece of graph paper, a...Ch. 8.3 - Random guesses (S). A multiple-choice test has 100...Ch. 8.3 - Random dates. There is a room filled with exotic...Ch. 8.3 - Random phones (H). Suppose you roll a 10-sided die...Ch. 8.3 - Fourever (ExH). Suppose you roll a die repeatedly,...Ch. 8.3 - Pick a number, revisited. In Mindscape 6, did you...Ch. 8.3 - Good start (H). Suppose the monkey is typing using...Ch. 8.3 - Even moves. Suppose you embark on a random walk on...Ch. 8.3 - Playing the numbers. Here is a numbers game. You...Ch. 8.3 - Random results. Someone looks at a list of 10,000...Ch. 8.3 - Monkey names. Suppose we have a monkey typing on a...Ch. 8.3 - The streak. Suppose you flip a fair coin 10 times...Ch. 8.3 - Girl, Girl,... (S). A couple has eight children....Ch. 8.3 - One mistake is okay. Suppose we try the IBM...Ch. 8.3 - Picking and matching. You and a friend...Ch. 8.3 - Picking and matching. You and a friend...Ch. 8.3 - Dice are different (ExH). In Mindscape 14 you...Ch. 8.3 - Death row (H). You may have noticed that two pairs...Ch. 8.3 - Striking again. Consider the striking deal game...Ch. 8.3 - Random returns. Suppose we take a random walk on...Ch. 8.3 - Random natural. Suppose you have a 10-sided die...Ch. 8.3 - Ace of spades. You randomly shuffle a deck of...Ch. 8.3 - Flipping out. You flip a coin eight times and get...Ch. 8.3 - Random reigns (H). Your (very egalitarian) school...Ch. 8.3 - Survey says... A survey in your intro psychology...Ch. 8.3 - Rock n roll. As a project in your Geology and Art...Ch. 8.3 - Random pitch. During a practice game, your...Ch. 8.4 - Have a heart. If you draw a card from a regular...Ch. 8.4 - Have a head. You ffip a coin 10 times. What is the...Ch. 8.4 - Sunny surprise. Suppose the chances are 1 in 2...Ch. 8.4 - Elephant ears. Suppose a quarter of all the...Ch. 8.4 - Little deal. In how many ways can you select three...Ch. 8.4 - The gym lock. A lock has a disk with 36 numbers...Ch. 8.4 - The dorm door. A dormitory has an electronic lock....Ch. 8.4 - 28 cents. How many different ways can you make 28...Ch. 8.4 - 82 cents. How many different ways can you make 82...Ch. 8.4 - Number please. Someone you really wanted to go out...Ch. 8.4 - Dealing with jack. Suppose you deal three cards...Ch. 8.4 - MA Lotto (H). To win the jackpot of the...Ch. 8.4 - NY Lotto (H). To win the jackpot of the New York...Ch. 8.4 - OR Lotto. To win the jackpot of the Oregon lottery...Ch. 8.4 - Burger King (S). You take a summer job making...Ch. 8.4 - More burgers. Suppose you are working at a burger...Ch. 8.4 - NetFlix. You have 65 movies on your NetFlix queue....Ch. 8.4 - Cineplex (ExH). Your local Cineplex is showing...Ch. 8.4 - One die. You roll a fair die four times. What is...Ch. 8.4 - Dressing for success. You have 5 T-shirts, 10...Ch. 8.4 - Band stand. The Drew Aderburg Band is planning a...Ch. 8.4 - Mondays undies. You are spending the weekend at a...Ch. 8.4 - Prob. 23MSCh. 8.4 - Cranking tunes. Your car stereo can be programmed...Ch. 8.4 - The Great Books. There are 20 Great Books, from...Ch. 8.4 - Morning variety (S). You wish to have a different...Ch. 8.4 - Blind mans bluff. You and a friend each pick a...Ch. 8.4 - Crime story (ExH). Suppose 20 witnesses saw...Ch. 8.4 - Theres a 4 (H). Someone you really want to go out...Ch. 8.4 - Making up the test. Your math prof says there will...Ch. 8.4 - Moving up. You have a part4ime job in a department...Ch. 8.4 - Counterfeit bills. You are given ten $100 bills...Ch. 8.4 - Car care. A burglar wishes to break into a car...Ch. 8.4 - Coins count. On your bureau you have a half...Ch. 8.4 - Math mania. There are 90 students enrolled in Math...Ch. 8.4 - Party on. You want to throw a party and can invite...Ch. 8.4 - No dice (H). You roll a pair of dice 24 times....Ch. 8.4 - Three angles. Draw 10 points on a piece of paper...Ch. 8.4 - Four parties (ExH). You want to have a party and...Ch. 8.4 - Making the cut. In 1988, the ignition keys for...Ch. 8.4 - Parental pride. The number of pictures you have of...Ch. 8.4 - Sister, Sister. The number of students in a...Ch. 8.4 - In search of a cubic. This section explains that...Ch. 8.4 - Cancellation! (H) This section explains that the...Ch. 8.4 - Keep on cancellin! Building on the previous...Ch. 8.5 - No pop quizzes. Your instructor gives mom quizzes...Ch. 8.5 - No easy quizzes (S). Your instructor starts giving...Ch. 8.5 - Silly sickness. Based on your symptoms, your...Ch. 8.5 - Making an algebra-down. Consider the equation...Ch. 8.5 - A beautiful mind. Who was John Nash? Give as many...Ch. 8.5 - Upon further study. On your instructor quizzes,...Ch. 8.5 - Power pass (ExH). The matrix below gives the yards...Ch. 8.5 - Pass with a p. Suppose the defense referred to in...Ch. 8.5 - Random run. Suppose now in Mindscape 7 the defense...Ch. 8.5 - Another random nm. Suppose now the defense...Ch. 8.5 - Lining up the defense (S). Using the payoff matrix...Ch. 8.5 - De-lines of defense. Using the payoff matrix in...Ch. 8.5 - Approximate Nash (ExH). Take the two lines from...Ch. 8.5 - Precise Nash. Use your answers to Mindscapes 8 and...Ch. 8.5 - Positive payoof (H). Below is a payoff matrix for...Ch. 8.5 - Estimating the equilibrium. Estimate the...Ch. 8.5 - Exacting the equilibrium. Find equations for the...Ch. 8.5 - Payoffs need not be positive. Heres a payoff...Ch. 8.5 - Negative equilibrium. Find equations for the two...Ch. 8.5 - Making negative sense. In what kind of game or...Ch. 8.5 - Colin and Tubbes. When first grader Cohn comes...Ch. 8.5 - Cohn and Tubbes 50—50. Suppose in Mindscape 21...Ch. 8.5 - Averaging Cohn and Tubbes. Suppose p is the...Ch. 8.5 - Colin and Tubbes line up (ExH). Referring to...Ch. 8.5 - Colin and Tubbes with a dash of Nash (S). Find the...Ch. 8.5 - Diagnosis. Based on your symptoms, your doctor...Ch. 8.5 - Sploosh test (H). Given the scenario in the...Ch. 8.5 - Diagnose the data. Using the data from the...Ch. 8.5 - My theory ... You are a paleontologist who finds...Ch. 8.5 - Going stag. Two Englishmen, Neville and Winston,...Ch. 8.5 - Selling sweets. Two snack cake companies, Snacky...Ch. 8.5 - Prisoners dilemma. Two people are suspected of...Ch. 8.5 - Match point. Find the coordinates (p,y) where the...Ch. 8.5 - Match lines. Find an equation for the line passing...Ch. 8.5 - Unmatched slopes. Two lines intersect at the point...Ch. 8.5 - Plotting percentages. Of the students in your...Ch. 8.5 - Itching for x (H). You go to the Health Center...
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