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The surn of the rank and nullity of a matrix equals the

Elementary Linear Algebra: A Matrix Approach | 2nd Edition | ISBN: 9780131871410 | Authors: Lawrence E. Spence ISBN: 9780131871410 187

Solution for problem 68 Chapter 1.4

Elementary Linear Algebra: A Matrix Approach | 2nd Edition

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Elementary Linear Algebra: A Matrix Approach | 2nd Edition | ISBN: 9780131871410 | Authors: Lawrence E. Spence

Elementary Linear Algebra: A Matrix Approach | 2nd Edition

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Problem 68

The surn of the rank and nullity of a matrix equals the number of rows in the matrix.

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AST Study Guide D long wavelengths. are more energetic than C the wavelengthe energy, the shorter (smaller) B When a photon comes in, you go up B the Earth’s electrons want to connect with C Blue=coming to you B Red shifted is positive Blue shifted is negative Longer=Redder D (Red/blue)oes not affect Doppler shift Know the order of the planets. Around 3 AU from the sun is the water boiling point. Protoplanetary disk rotating circumstellar disk of dense gas and dust surrounding a young newly formed star The nebular hypothesis of solar system formation describes how protoplanetary disks are thought to evolve into planetary systems. B the planet), short distanceily maneuvered by A Look at the location of both orbits. They are basically opposites. When the star is close to K, the planet is close to I. The question is asking where the planet would be located, when the star is redshifted. The star is redshifted at K, which means the planet is at I. 1. Radio velocity, transit method. 2. Transit Method. Radio velocity: A star with a planet will move in its own small orbit in response to the planet's gravity. This leads to variations in the speed with which the star moves toward or away from Earth, i.e. the variations are in the radial velocity of the star with respect to Earth. The radial velocity can be deduced from the displacement in the parent star's spectral lines due to the Doppler effect. The radial-velocity method measures these variations in order to confirm the presence of the planet. Direct Imaging: Only works for very nearby stars and planets with large orbits. Transit photometry: While the above methods provide information about a planet's mass, this photometric method can determine the radius of a planet. If a planet crosses (transits) in front of its parent star's disk, then the observed visual brightness of the star drops a small amount. C warmed by visible light.cked. The Earth is D I have no clue what this is D You see a thin sliver of Moon visible in the phase is the moon infore sunrise. What Waning Crescent You can fit two moon orbits in the sun. A The moon is in the way during a solar eclipse. Understand the parts of the sun and their processes. Core: Nuclear fusion takes place in the core. -Hydrogen(H) changes to Helium(He) - In this process, some of the mass of the hydrogen atoms is converted into energy in the form of light. Photosphere: A star’s outer shell. Sunlight comes from the Photosphere. Chromosphere: it can be seen only during a total eclipse, where its reddish color is revealed. -The temperature at the top of photosphere is only about 4,400 K, while at the top of reaches 25,000 K.1][This is however the opposite of what we find in the photosphere, where the temperature drops with increasing phenomenon causes the temperature of thehat chromosphere to paradoxically increase further from the Sun's interior. D The differences in the spectral lines in stars is mainly caused by their differences in: Temperature The temperature stretches the spectrum. B Absolute magnitude= 5 Apparent= 0 Example: The sun has an apparent magnitude of -26.7, and an absolute bright to us, and when placed at 10 parsecs away, it gets dimmer. This proves that the sun is closer to us than 10 parsecs. C The sun’s spectral type is G. B which is correct.e process of nuclear fusion, A Think of a Prius and a Hummer. A C The answer is C, unless stated on the test that the smaller object is blue. If the object is blue, it is brighter because what matters is the temperature (luminosity per unit area). Blue=Hotter than red. In that case the answer would be B. Understand the steps during the evolution of low mass stars . Stellar evolution tutorial. Understand the different paths that a star can follow in its evolution. Extra Info from review • The division between low mass and high mass stars is around 8 solar masses. • “Harvard Computers” • -Henrietta Swan Leavitt discovered thousands of variable stars, the Cepheid period-luminosity relation. The PL relation is important because it provides us with the absolute magnitude, which can give us the apparent magnitude, and ultimately allows us to determine the distance. Annie Jump Cannon is credited with the creation of the Harvard Classification Scheme. Anonia Maury built on that idea and suggested the classes be arranged by temperature; OBAFGKM. Cecilia Payne-Gaposchkin applied Meghnad Saha’s theory of ionization of atoms to explain the changing spectral lines in different classes as caused by temperature. • Stars, no mater what class, are mostly made of hydrogen. Williamina Fleming was instrumental in discovering white dwarf stars. YOU DO NOT NEED TO MEMORIZE NAMES, JUST UNDERSTAND THE ACHIEVEMENTS MADE AS A GROUP BY THE HARVARD COMPUTERS Important Lecture Slide Info (Star/Sun Slides) • The most abundant element in the Universe is hydrogen, made of a proton and electron. • The net result of nuclear fusion of 4 H nuclei is 1 He nucleus, neutrinos, and energy. • • 4 H nuclei is slightly heavier than 1 He, the remaining mass is released as energy. • With the force of gravity pointed toward the center, the density is the highest in the core. • Higher density means higher temperature and higher rate of nuclear fusion. Cont. • Sunspots gives us clues about its rotation; period is about 1 month. • It varies depending on the latitude (differential rotation). • Number of sunspots goes through 11-year cycles. • Strong magnetic field penetrates the photosphere and connects the corona to the interior via sunspots. • The Sun is extremely bright and gives off an immense amount of energy, but it turns out that it isn't even close to giving off a truly continuous spectrum. (all the way from red to violet, with no gaps) • Thus, emission spectra are produced by thin gases in which the atoms do not experience many collisions (because of the low density). The emission lines correspond to photons of discrete energies that are emitted when excited atomic states in the gas make transitions back to lower-lying levels. • A continuum spectrum results when the gas pressures are higher. Generally, solids, liquids, or dense gases emit light at all wavelengths when heated. • An absorption spectrum occurs when light passes through a cold, dilute gas and atoms in the gas absorb at characteristic frequencies; since the re-emitted light is unlikely to be emitted in the same direction as the absorbed photon, this gives rise to dark lines (absence of light) in the spectrum. What type of spectrumwould we observe, if we only look at light fromthe photosphere of the Sun • Continuous spectrum • • Absorption spectrum • • Emission spectrum • Absorption • What type of spectrum would we observe, if we only look at light from the chromosphere of the Sun • Continuous spectrum • • Absorption spectrum • • Emission spectrum • Emission • The Chromosphere can only be seen easily in a total solar eclipse. When it can be seen it is reddish in color. This color is the origin of its name (chromos meaning ``color''). • emitting at discrete wavelengths.e is due to an emission spectrum from hot, low density gases Cont. • H-R Diagram • Independently developed by Dane Ejnar Hurtzsprung and American Henry Norris Russell. • Stars spend most of their lives on the main sequence. • At this stage, stars consume hydrogen, and produce helium and photons. • Red Giants: Cold, Bright, Large • White Dwarves: Hot, Faint, Small • Spectroscopic parallax • A method to measure the distance of a star, but it has nothing to do with parallax Main sequence = mass sequence • Main sequence stars form a tight sequence in HR diagram from hot/bright to cold/red For main sequence stars, bluer stars are _________ than redder stars. • more massive and brighter • Most stellar (solar) systems have more than 1 star. • • Binary systems (2 stars) are the most common. • When light from both stars are unobstructed, we observe the most amount of light. • When the brighter star is behind, we observe the least amount of light. • • What if the smaller star is bluer What matters is the temperature (luminosity per unit area). Bluer=Hotter B. M67. The cluster with the moststars branching offis the oldest because a lot of the stars in the cluster have died and branched off. • • The formation of stars and planets is the same event. • • Stars and planets are formed by the gravitational collapse of clouds of dust/gas. • • The dense cool clouds mainly composed of hydrogen molecules are called “molecular clouds.” • • These molecular clouds are observed to be stellar nurseries. • Gravitational collapse is a “capitalistic” process. The rich gets richer. • • The parts of the cloud that have slightly more mass will attract more mass via gravity. • Infrared cuts through dust, much like radio cuts through walls. • Which planet formed closest to the Sun at temperatures below the freezing point of water • • Venus • • Earth • • Mars • • Jupiter • Radial velocity • • The star and the planet are always moving in opposite directions and on the opposite sides. • A large doppler shift is observed for a system with • • high mass planet, • • low mass star, • • small distance between star and planet. This is the opposite version of the question provided in the review. Theansweris K. When the planet is at K, the star is at I. this means the star would appear blueshifted as it orbits toward Earth. • Convection drives Oceanic and Continental Plates together, creating volcanoes and ocean trenches, erasing surface features. Without greenhouse effect, surface temperature of the Earth would be at -18C or 0F on average and have large variations. • More H2O and CO2 in the atmosphere strengthens the greenhouse effect. • Venus is hotter than Mercury, even though it is twice as far away from the Sun. • Which type of light do greenhouse gasses absorb/emit • • Infrared • • Visible • • Ultraviolet • Greenhouse effect: CO2 from vehicles and industry is emitted to the atmosphere. This creates a layer round the earth that traps heat from the sun. This is similar to the glass in a greenhouse trapping the heat inside. This is why its called the 'greenhouse effect', and why CO2 is known as a greenhouse gas. • Ozone Depletion: The ozone layer around the earth protects the earth from UV rays from the sun. The ozone layer is destroyed by chemicals known as CFC's. These were found in coolant in old fridges and aerosols. Since CFC's were banned in the 1980's the ozone layer has began to heal itself. • Global Warming: the recent increase in the world's temperature that is believed to be caused by the increase of certain gases (such as carbon dioxide) in the atmosphere. Waning is the bottom half. Waxing is the top half. Gibbous is on the left Crescent is on the right. First quarter is on top Third Quarter on bottom. (Think third is on bottom because three comes after one.) The moon works the same as the sun. It rises in the East, sets in the West. It is highest in the South. • In the Northern Hemisphere, when the Moon is high up in the sky, it is toward the • • North • • South • • East • • West After labelling this picture, you can determine when the moon rises/sets for each phase. For the Waxing crescent phase the moon is highest at 3pm. Simply subtract 6 hours to find when it rises (9am) and add 6 hours to find when it sets(9pm). For the Waning crescent, it is highest at 9am. Again, subtract 6 hours to find when it rises(3am) and add 6 hours to find when it sets(3pm).

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Chapter 1.4, Problem 68 is Solved
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Textbook: Elementary Linear Algebra: A Matrix Approach
Edition: 2
Author: Lawrence E. Spence
ISBN: 9780131871410

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