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Solved: [M] In parts (a)–(d), let U be the matrix formed

Linear Algebra and Its Applications | 4th Edition | ISBN: 9780321385178 | Authors: David C. Lay ISBN: 9780321385178 62

Solution for problem 36E Chapter 6.2

Linear Algebra and Its Applications | 4th Edition

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Linear Algebra and Its Applications | 4th Edition | ISBN: 9780321385178 | Authors: David C. Lay

Linear Algebra and Its Applications | 4th Edition

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Problem 36E

[M] In parts (a)–(d), let U be the matrix formed by normalizing each column of the matrix A in Exercise 35. Reference:[M] Show that the columns of the matrix A are orthogonal by making an appropriate matrix calculation. State the calculation you use.

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Chapter 13: Learning-Strategy Training, Cooperative Learning, and Multiple Intelligences  The three divisions of this chapter deal with: (a) Identifying the traits that characterize a successful learner and (b) Helping the student learn how to study, when to study, and under what conditions to study.  This material is not specifically geared toward second-language acquisition; instead, it is applicable to studying in general, for any subject. o These details are well worth considering as you attempt to help and encourage each individual student to become a successful language learner. o However, the material on “Multiple Intelligences” is sheer crap.  As we’ll see when we get to that section, there is only one form of intelligence: intelligence.  Calling the ability to stack cups rapidly “intelligence” is like calling a corndog a Porterhouse steak: you can call it that if you wish, but that doesn’t make it one.  Learning-Strategy Training: o The characteristics of successful learners are worth noting:  They are willing and accurate guessers, who have a strong desire to communicate and will attempt to do so even at the risk of appearing foolish. They attend to both the meaning and the form of their message. They also practice and monitor their own speech as well as the speech of others. (p. 159)  Let’s consider these traits individually.  First, successful learners –not just in language but in any field –are risk takers:  That is, they are willing to form and test a hypothesis about the subject matter –in our case, speaking the target language.  In order to do so, they will endeavor to speak the target language in order to communicate with a native speaker or with another person who shares the target language. (1) Again because such learners are risk takers, they are highly motivated to communicate, despite any concerns they might have about sounding foolish by making mistakes in vocabulary, grammar, or idiom: they inherently emphasize that if they wait until they can speak the TL perfectly, they will miss many opportunities to communicate. o If they make mistakes, they will either self-correct or accept the correction of a native speaker  Second, successful learners pay attention to the two basic components of what they’re trying to say –i. e., the meaning of the words (semantic content) and the form in which they express it (structure of the language). o No one questions the importance of semantic content:  The words in a sentence must add up to the speaker’s intended meaning.  Structurally, languages tend to fall into one of two basic types: o Synthetic, in which the meaning of the words in a sentence is expressed by their inflections, and o Analytical, in which the positions of the words in a sentence determine their meaning.  Old English (Anglo-Saxon) is a Synthetic language, with essentially the same inflections as modern German; however, over the course of its long evolutionary history, English has become primarily an Analytical language, although it retains certain inflections.  Obviously, in Synthetic languages, form is of paramount importance, whereas  In Analytical languages, structure holds the key to meaning. o Successful language learners grok the importance of both semantic content and structure in the TL.  Finally, successful language learners practice the TL –in the language lab, at home, in conversations with their pets, before a mirror, into a recorder, in the shower. ..The list goes on... o Having a conversation partner who is fluent in the TL helps, but the absence of one will not stop a determined L2 learner. o He/She will listen to the TL as he/she speaks it, as others (especially native- speakers) speak it, and –an excellent technique –as native-speaking actors or newscasters speak it in motion pictures or television programs.  As we’ve discussed before, demonstrating conversational competence constitutes more than merely having a large vocabulary and being able to speak the TL grammatically:  It includes speaking in the characteristic rhythms, perhaps employing some of the (socially acceptable) body language used in the TL culture  And knowing a goodly number of idiomatic expressions and how to use them correctly. o Watching films–particularly with the English subtitles turned on -can assist the L2 learner in acquiring these characteristics of the TL. o But whether he/she is watching a film or conversing with a native speaker of the TL, a successful language learner is listening carefully and storing away information for the next conversation.  The Experience under “Learning-Strategy Training” includes the teacher’s reading of the students “learning journals” –what he calls “Language Logs.” o Consider requiring the students to keep a Language Log to be an excellent practice:  Asking them to collect one new word or expression per week (or per day or per month, at your preference) makes them more aware of the language being spoken around them. o Encourage students to listen to the conversations of people in stores, restaurants, airports, et al., and record any particularly intriguing expressions in the Language Logs.  Television programs, motion pictures, magazines–virtually any media – can be useful sources of entries.  Ask students to cite the place where they encountered the term, the time, and other details of the context (such as who was speaking); they must not only record the expression but also explain it. o If you assign a Language Log, you must read your students’ logs regularly:  This helps you to monitor the kind of material they are collecting and will also give them a sense that you appreciate their efforts.  Discourage students from merely going to a dictionary or website and copying down words and their definitions:  This is sheer laziness  The explanations of strategies for deriving meaning through drawing inferences from captioned photographs, highlighting article titles and subheadings, and skimming for key words are excellent, and these techniques are effective: o He highly recommend exercises such as those described in the lesson.  These will enhance your L2 students’ ability to read and understand newspaper and magazine articles, both in conventional print format and online.  Notice that at one point in the experience, a student asks the teacher what “endangered” means, and the teacher encourages him to guess –but directs him toward the part of the word that he might understand (“danger”). o Hence, the teacher is not promoting wild guessing at the meaning of an unknown term; instead, he is showing the student how to analyze a word systematically.  The Multifarious Mysteries of Metacognitive Management o The term “metacognitive strategies” translates as “thinking about thinking about studying.” He thinks.  A less obfuscatory way of saying the same thing would be “planning how to maximize the effectiveness of your study sessions.”  Hence, a student who knows what kind of environment he/she needs in order to study effectively (e.g., quiet, solitary), knows when during the day he/she studies best (morning, afternoon, evening), and how long he/she should allow him-/herself to study (typically approximately one hour per subject) can choose the conditions that will enable him/her to get the greatest benefit out of studying. o To assist your students in determining these factors for themselves, you may wish to have them answer a brief questionnaire and inventory: for example:  If a student studies best in a quiet environment, where at home or at school or in some other accessible place (e. g., the public library) can he/she find such a location  If a student works best with a study partner fluent in the TL, how can he/she locate such a partner  The other study strategies discussed on p. 164 are also excellent and are worth bringing to your students’ attention.  Cooperative (or Collaborative) Learning o We’ve actually been discussing various forms of cooperative, or collaborative, learning(CL) since the beginning of the term:  Any time you have students work in groups or teams, any time the class as a whole generates a dialogue or prepares a skit, any time you assign your students a task which five or six students must collaborate on, they are engaging in cooperative learning.  An essential rule of working collaboratively is that the students must carry on all of their discussions in English. o In addition to practicing the TL and accomplishing the task, the CL Method encourages students to use various social skills to encourage and exhort each other –everything from “Way to go!” and “Attaboy!” to smiles and hugs or pats on the back. o (Make sure to check your school’s policies about physical contact: many have a sort of “catch-and-release” rule to avoid excessive or inappropriate physical contact.  Multiple Intelligences o There is no possible disputing the fact that individuals demonstrate different learning/cognitive styles.  Some students are visual learners, some aural learners, while some are successful vicarious learners.  Hatch typifies some learners as data-gatherers while others are rule-formers o These classifications appear to mirror the classical rhetorical styles of inductive reasoning and deductive reasoning, respectively. o In general, data-gatherers tend to be more fluent in the TL but may demonstrate inaccuracies in vocabulary, grammar, and/or structure o Conversely, rule-formers follow the learned rules accurately, but tend to be less fluent for fear of violating the rules.  Of course, it is possible for a learner to demonstrate a synthesis of both modes. o In real language, “multiple intelligences” may be called “aptitudes” or “skills.” o He doesn’t wish to diminish the importance of aptitudes in learning or in performing various useful, necessary tasks; however, I strongly believe that taking a term like “intelligence” and proposing that it also mean expertise in basket weaving is to add massive confusion to the language.  Educators are justifiably skeptical of IQ tests, which may or may not measure intelligence; however  They do measure something (expressed as “G” by psychologists), which has proven to be the best overall predictor of success in any given field. o A good working definition of “intelligence” for our purposes is “the capacity of an individual to successfully adapt to his/her environment or adapt the environment to fit him-/herself.”  This definition goes a long way toward explaining why IQ tests are typically heavily weighted toward verbal abilities:  as a general rule o Fluency in the dominant language of a culture constitutes a major component in successful adaptation to the environment.  An awareness of your students’ learning styles and array of aptitudes can help you, as an ESL teacher, to adapt various exercises and tasks to the students’ abilities. o Doing so will help you maintain their interest and offer them greater opportunities for success both in the classroom and outside it. (1) Sometimes this will prove to be the case in an ESL class. A boy whose native language is Latvian and a girl whose native language is Tagalog will, perforce, need to speak English in order to be mutually understood. (2) Strategies for encouraging students to use only English include both rewards and “fines.”  For example, have the students collaboratively design a form of classroom currency that can be printed out and distributed to everyone in the class. o At the beginning of each week, give each student ten classroom “dollars”; then, for each violation of the “English-only” rule, fine him/her one dollar.  Students can also earn extra dollars for volunteering to answer difficult questions and giving a correct answer, or for successfully completing some task.  At the end of the week, allow the students to redeem their classroom currency for candy or pencils or some other treat, or they can save their money until they have saved up enough to buy something more impressive.  Perhaps you could even have the class pool their hoarded currency and throw them a pizza party (if your school allows it) once they’ve reached a certain amount.

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Chapter 6.2, Problem 36E is Solved
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Textbook: Linear Algebra and Its Applications
Edition: 4
Author: David C. Lay
ISBN: 9780321385178

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Solved: [M] In parts (a)–(d), let U be the matrix formed