# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

38 Pages

### ch6

Course: MATH 21B, Summer 2008
School: Harvard
Rating:

Word Count: 13123

#### Document Preview

6 ISM: Chapter Linear Algebra Chapter 6 6.1 1. Fails to be invertible; since det 2. Invertible; since det 3. Invertible; since det 1 2 = 6 - 6 = 0. 3 6 2 3 = 10 - 12 = -2. 4 5 3 5 = 33 - 35 = -2. 7 11 1 4 = 8 - 8 = 0. 2 8 4. Fails to be invertible; since det 2 5. Invertible; since det 0 0 6 6. Invertible; since det 5 3 7. This matrix is clearly not invertible, so the determinant must be zero. 5 7 11 7 =...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Massachusetts >> Harvard >> MATH 21B

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Harvard - MATH - 21B
Chapter 7ISM: Linear AlgebraChapter 7 7.11. If v is an eigenvector of A, then Av = v. Hence A3 v = A2 (Av) = A2 (v) = A(Av) = A(Av) = A(2 v) = 2 Av = 3 v, so v is an eigenvector of A3 with eigenvalue 3 .1 2. We know Av = v so v = A-1 Av = A-1 v
Harvard - MATH - 21B
ISM: Linear AlgebraSection 8.1Chapter 8 8.11. e1 , e2 is an orthonormal eigenbasis. 2. 3.1 21 1 , 12 1 -1is an orthonormal eigenbasis.2 -1 , 15 is an orthonormal eigenbasis. 1 2 1 1 1 1 1 1 4. 3 1 , 2 -1 , 6 1 is an orthono
Harvard - MATH - 21B
Chapter 9ISM: Linear AlgebraChapter 9 9.11. x(t) = 7e5t , by Fact 9.1.1. 2. x(t) = -e e-0.71t = -e1-0.71t , by Fact 9.1.1. 3. P (t) = 7e0.03t , by Fact 9.1.1. 4. This is just an antiderivative problem: y(t) = 0.8 t2 + C = 0.4t2 + C, and C = -0.
Harvard - MATH - 21B
Chapter 5ISM: Linear AlgebraChapter 5 5.1 72 + 112 = 49 + 121 = 170 13.04 2. v = 22 + 32 + 42 = 4 + 9 + 16 = 29 5.39 3. v = 22 + 32 + 42 + 52 = 4 + 9 + 16 + 25 = 54 7.35 1. v = 4. = arccos 5. = arccos 6. = arccosuv u v uv u v uv u
Harvard - MATH - 21B
ISM: Linear AlgebraSection 4.1Chapter 4 4.11. Not a subspace since it does not contain the neutral element, that is, the function f (t) = 0, for all t. 2. This subset V is a subspace of P2 : The neutral element f (t) = 0 (for all t) is in V .
Harvard - MATH - 21B
ISM: Linear AlgebraSection 3.1Chapter 3 3.11. Find all x such that Ax = 0: . 1 2. 0 . . . 0 3 4. - . 1 0. 0 , so that x = x = 0. . 1 2 . . 0 0 1.ker(A) = {0}. 2. Find all x such that Ax = 0: . . 3. 2 3. 0 - 1 2 . 0 , so that . . . 6 9. 0 .
Harvard - MATH - 21B
Chapter 2ISM: Linear AlgebraChapter 2 2.11. Not a linear transformation, 0 2 2. Linear, with matrix 0 0 1 0 9 3 -3 1 2 -9 4. A = 4 -9 -2 5 1 5 5. By Fact 2.1.2, the three columns of the 2 3 matrix A are T (e1 ), T (e2 ), and T (e3 ), so
Harvard - MATH - 21B
ISM: Linear AlgebraSection 1.1Chapter 1 1.11. x + 2y x + 2y = 1 -y 2x + 3y = 1 -2 1st equation x + 2y = 1 -2 2nd equation x y=1 y 2. 4x + 3y 7x + 5y3 x + 4y 1 -4y 3 x + 4y = 2 4 =3 7x + 5y=1 = -1 (-1)= -1 , so that (x, y) = (-1, 1). =
UGA - ACCT - 7605
ACCT 7605 Exam I- Practice Questions Gleim's Exam Questions Test Prep Financial Accounting [1] Gleim #: 25.1.1 - Source: Publisher A business combination may be structured as a merger, a transfer of equity interests, a direct acquisition of net asset
UGA - LEGL - 4400
Chapter 14: Mistakes, Fraud, and Voluntary Consent 1) Mistakes of Fact: occur in two forms bilateral and unilateral a) Bilateral Mistake made by both of the contracting parties; occurs when both parties are mistaken as to an existing material fact
UGA - LEGL - 4400
Chapter 11: Agreement 1) Requirements of the Offer: a) Offer: is a promise or commitment to do or refrain from doing some specified thing in the future. b) 3 elements necessary for an offer to be effective: i) The Offeror must have a serious intentio
UGA - LEGL - 4400
Chapter 15: The Statute of Frauds Writing Requirement 1) The Origins of the Statute of Frauds a) In 1677, statute passed by the English Parliament known as the Act for the Prevention of Frauds and Perjuries i) Established that certain types of contr
UGA - LEGL - 4400
Chapter 16: Third Party Rights 1. Privity of Contract: a contract that is a private agreement between the parties who have entered into it, and traditionally these parties alone have rights and liabilities under the contract. a. Third party: one who
UGA - LEGL - 4400
Chapter 13: Capacity and Legality 1) Contractual Capacity: the legal ability to enter into a contractual relationship. a) Minors i) The age of majority for contractual purposes is eighteen years ii) Minority status may be terminated based on marriage
UGA - LEGL - 4400
Chapter 12: Consideration 1) Consideration: defined as the value given in return for a promise or in return for a performance. a) Is broken down into two parts: i) Something of legally sufficient value must be given in exchange for the promise, and i
UGA - LEGL - 4400
Chapter 10: Contract Law Nature and Terminology 1) Function of Contract Law: i) Provides an essential condition for the existence of a market economy. ii) Without a legal framework of reasonable assured expectations within which to make long-run plan
UGA - ACCT - 7605
ACCT 7605 Exam II-Practice Questions Gleim's Exam Questions Test Prep Financial Accounting [Fact Pattern #1]Parma Corp. and Seville Corp. condensed balance sheets on January 1, year 1 are presented below. On January 2, year 1, Parma borrowed \$60,000
UGA - ACCT - 7651
Forensic Accounting &amp; Fraud Examination Week One Lecture NotesFORENSIC ACCOUNTING: - the big picture idea - the use of accounting knowledge or skill for courtroom purposes; Forensic accounting can therefore involve not only fraud, but also bankrupt
Michigan State University - SOC - 100
Sociology the scientific study of social behavior and human groups. Sociological imagination - awareness of the relationship between the individual and the wider society. Theory used to explain problems, actions, or behavior Verstehen insight, in
Michigan State University - HB - 210
Exam 1 What state has the highest tax rate for gambling? Illinois What is the Detroit tax rate? 24% Only city in Nevada that doesnt allow gaming? What do pit bosses do, dealers? Title 31 Payouts for craps Myths about slot Which states dont allow gamb
Michigan State University - GEO - 221
John Russo A38669634 GEO221 September 8, 2008 Homework #2 Figure 1: 1) 2) 3) 4) 5) Figure 2: 1) 2) 3) 4) 5) Cylindrical Equatorial Conformal Preserves shape Mercator Azimuthal Equatorial equivalent Preserves area Albers Equal Area
Michigan State University - GEO - 221
John Russo A38669634 Assignment #1 September 1, 2008Geographic information is the knowledge and collection of geographical features. Knowledge of these features allows us to be more efficient and precise in navigation, calculations, census data, te
Michigan State University - ECON - 301
John Russo EC301 September 8, 2008 Homework #1 1) a) 208.8100*1.88=3.93 The real price has decreased from \$3.93 in 1980 to \$2.52 in 2000. b) 2.52-3.933.93= -35.9% The percentage change in real price from 1980 to 2000 is a decrease of 35.9%. 2)3)L
UIllinois - MATH - 241
UIllinois - MATH - 241
UIllinois - MATH - 241
UIllinois - MSE - 182
UIllinois - MSE - 182
UIllinois - MSE - 182
North Texas - BIOL - 1720
Dynamics of EcosystemsAn ecosystem includes all the organisms that live in a particular place plus the abiotic environments in which they live1) Energy constantly enters and is lost. 2) Chemicals move through biogeochemical cyclesBIOGEOCHEMICAL
North Texas - BIOL - 1720
Chapter 52 Behavioral BiologyInnate behavior Learning and cognition Behavior and Communication Evolution shapes behavior Social BehaviorBehavior: How an animal responds to environmental stimuliApproaches to behavior science Proximate causation (H
North Texas - BIOL - 1720
POPULATION ECOLOGYEcology: the study of how organisms relate to one another and to their environments. Physical environment (abiotic factors): Temperature, Water, Sunlight, SoilHomeostasis: plants and animals maintain an internal environment Confo
North Texas - BIOL - 1722
Exam 9 Chapter 56 Chapter 57
North Texas - BIOL - 1720
1. Ultimate 2. ecology 3. optimal foraging 4. reproduction 5. learning 6. mate choice 7. kin selection 8. e 9. c 10. c 11. c 12. e 13. b 14. c 15. d 16. e 17. a 18. 1-a, 2-c, 3-b, 4-e, 5-d 19. Biotic potential 20. carrying capacity 21. r 22. K 23. de
North Texas - BIOL - 1720
Biology 1720, Fourth Non-Comprehensive Exam, 2005 1. Which of the following is NOT a factor in the opening of the stomata? a. a decrease in turgor pressure in the guard cells b. movement of potassium ions (K+) into the guard cells c. light d. the inf
North Texas - BIOL - 1720
BIOLOGY 1720/1722 FINAL EXAM, MAY, 2004 Please read each question carefully and choose THE BEST answer. Please choose only ONE answer (a,b,c,d, or e). Please complete the exam in the time allotted (50 minutes). 1. The phytochrome system activates cer
North Texas - BIOL - 1720
Chapter 551. 2. 3. 4. 5. 6. 7. A(n) _ cycle is a cyclic path that involves both biological and chemical processes of an ecosystem. A(n) _ is a permeable underground layer of rock, sand, and gravel saturated with groundwater. The upper, unconfined po
North Texas - BIOL - 1720
Chapter 541. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. All organisms living together in a place are called a _. Competition shapes communities because there are usually limited _. The attempt of two organisms trying to utilize the same resource is called
North Texas - BIOL - 1720
Chapter 531. Populations of mammals that live in colder climates tend to have shorter ears and limbs than populations of the same species in warm climates (coyotes are a good example of this). This is called _ Rule. Populations of mammals that live
North Texas - BIOL - 1720
Chapter 521. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. When we ask why a behavior evolved or what its adaptive value is, this is a question of _ causation. _ studies behavior, behavioral genetics, and comparative psychology. Tinbergen is credited with foundin
North Texas - BIOL - 1720
Biology Test #4 1. In an ecosystem , which statement reflects the relationship w/ energy and nutrients o Energy flows and nutrients cycle in an ecosystem 2. Adult males are most likely to display monogamy if o a small # of vulnerable offspring result
North Texas - BIOL - 1720
Chapter 411. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Many plants can _ themselves by asexual reproduction. Asexually produced progeny are genetically _ to the parent individual. Flowering plants use animals or wind to transfer _ between flowers. Woody plant
North Texas - BIOL - 1720
Chapter 41 1. clone 2. identical 3. pollen 4. perennial 5. autonomous 6. annual 7. wind 8. outcrossing 9. abscission 10. endosperm 11. protoplast Chapter 52 1. ultimate 2. Ethology 3. ecology 4. optimal foraging 5. reproduction 6. learning 7. instinc
North Texas - BIOL - 1720
Chapter 52 1) When we ask why a behavior evolved or what its adaptive value is, this is a question of_ causation. 2) The field of behavioral _ studies how natural selection shapes behavior. 3) Animals tend to feed on prey that maximize their net ener
North Texas - BIOL - 1720
Community EcologyAll the organisms that live together at a time and in a place Individual Concept = aggregate of species Holistic Concept = integrated unitInteractions among species shape the nicheNiche = the total ways in which an organism uses
North Texas - ENGL - 1325
Manchanda 1 Ramneek Manchanda Writing About Literature II Allegra Davis 08 March 2008 Joy Luck Club Essay The Joy Luck Club is a series of sixteen interwoven stories about Chinese immigrant mothers and the American influenced daughters as theyre rela
North Texas - ENGL - 1325
Manchanda 1 Ramneek Manchanda Writing About Literature II Allegra Davis 29 April 2008 The Misfortunes of Time Travel Dana is a middle aged black woman living in California celebrating her twenty seventh birthday with her husband, Kevin, when he state
North Texas - ENGL - 1325
Manchanda 1 Ramneek Manchanda Writing About Literature II Allegra Davis 13 April 2008 Challenges of Human Interaction in the Future After civilization is on the brink of elimination due to a deadly nuclear war known as Flame Deluge, to specialty of h
North Texas - BIOL - 1722
EXAM 6 (Semester II)Chapter 47The Endocrine System The tissues and organs of the vertebrate body cooperate to maintain homeostasis (constant internal conditions) of the bodys internal environment and to control other body functions, such as reprodu
North Texas - BIOL - 1722
Conservation BiologyChapter 591Factors Responsible Causes of extinction: direct or indirect Overexploitation Habitat loss Introduced species Disruption of ecosystem interactions Pollution Loss of genetic variation Catastrophic disturbanc
North Texas - BIOL - 1722
Community EcologyChapter 561Biological Communities Community: all the organisms that live together in a specific place Evolve together Forage together Compete Cooperate2Biological Communities Communities can be characterized either by
North Texas - ENGL - 1325
Manchanda 1 Ramneek Manchanda Writing About Literature II Allegra Davis 30 March 2008 Abandoned Communication Near the Californian coast, the narrator decides to visit her expiring friend in the hospital for the first and last time. Fear overwhelms t
North Texas - BIOL - 1722
Exam 7Chapter 25Systematics and the Phylogenetic Revolution All organisms share many biological characteristics. They are composed of one or more cells, carry out metabolism and transfer energy with ATP, and encode hereditary information in DNA. Yet
North Texas - BIOL - 1722
Exam 8Chapter 52Behavioral biology Many behavioral patterns are innate o Approaches to the study of behavior Behavior can be defined as the way an animal responds to stimuli in its environment Proximate causation defines how something works, whil
North Texas - BIOL - 1722
Population EcologyChapter 551Environmental Challenge Ecology: the study of how organisms relate to one another and to their environments Abiotic: non-living; wind, temperature Key elements of the environment: Temperature Water Sunlight So
North Texas - BIOL - 1740
Presentation By: Ramneek Manchanda, Keshav Poddar, David Mathai, Nikhil KogantiFly PollinationBiotic Pollination 200,000 biotic pollinators Entomophily Flies, bees, ants, butterflies Brightly colored Scented Nectar Pollen grains Large O
North Texas - BIOL - 1740
Ramneek Manchanda Quinidine/Quinine In the modern Technological Era, we rely heavily on information about current sciences that can lead to key innovations in the pharmaceutical, medical and engineering field. The growth of the pharmaceutical industr
North Texas - BIOL - 1740
Ramneek Manchanda Evolution and Tinkering Analysis In the new Technological Era today, many issues have created much controversy among different extremist groups. One issue surrounding modern scientists and religion extremists is the theory of evolut
North Texas - PHYS - 1710
Physics Spreadsheet 24By: Robert Rogers Question 1: Enter the molar mass of O 2 (g/mol): Enter the molar mass of He (g/mol): Enter the rms speed of He atom (m/s): Molar mass of O 2 (kg/mol): Molar mass of He (kg/mol): rms speed of O 2 (m/s): Questio
North Texas - PHYS - 1710
manchanda (rm34538) Physics 1710 Homework 1 Weathers (17104) This print-out should have 9 questions. Multiple-choice questions may continue on the next column or page nd all choices before answering. 001 10.0 points The mass of the planet Saturn
North Texas - PHYS - 1710
manchanda (rm34538) Homework 2 Weathers (17104) This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page nd all choices before answering. 001 (part 1 of 2) 10.0 points The velocity of a particle m