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compute the rank and nullity of the given matrices over the indicated p

Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition | ISBN: 9780538735452 | Authors: David Poole ISBN: 9780538735452 298

Solution for problem 3.5.54 Chapter 3

Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition

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Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition | ISBN: 9780538735452 | Authors: David Poole

Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition

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Problem 3.5.54

compute the rank and nullity of the given matrices over the indicated p.

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Biol 213 – Exam no1 Study Guide (2/2) Chapter 3 3.1. What Kinds of Molecules Characterize Living Things  Proteins: formed from ≠ combinations of 20 common amino acids  Carbohydrates: sugar monomers (monosaccharides) linked together to form polysaccharides.  Nucleic acids (DNA & RNA): formed from 4 kinds of nucleotide monomers  Lipids: noncovalent forces maintain the interactions |b| the lipid monomers These biological molecules, characteristic of living organisms, are polymers –with the exception of lipids. Although lipids are not polymers, they are considered a special type of macromolecule. A macromolecule’s functional groups determine how it will function and interact with other molecules. Some functional groups and their chemical properties:  Hydroxyl (-OH): polar, forms hydrogen bonds with water to help dissolve molecules; alcohol.  Carboxyl (-COOH): polar and acidic, ionizes in living tissues to form –COO and H ; + carboxylic acids.  Aldehyde (-COH): aldehydes.  Keto (-C=O): the C=O group is very reactive, important in carbohydrates, in building molecules and in energy-releasing reactions; ketones.  Amino (-NH ): 2asic; amines.  Phosphate (PO ): 4cidic; when bonded to another phosphate, hydrolysis releases much energy; organic phosphates.  Sulfhydryl (-SH): by giving up H, two –SH groups can react to form a disulfide bridge, thus stabilizing the protein structure; thiols.  Ester (-COO): The carbon atom forms a double bond with one oxygen atom and a single one with the other. Types of Isomers: - Structural: their atoms are joined together differently. - Cis-trans: different orientation around a double bond (cis –same side, trans –opposite sides) - Optical: they occur when an asymmetric carbon has 4 ≠ atoms or groups attached to it. Optical isomers are mirror images/reflections of one another. N.B: Some biochemical molecules that can interact with one optical isomer are unable to “fit” the other. Biochemical unity is in conformity with the belief that there is a common ancestor to all living organisms. Proteins make up more than half of our macromolecules, followed by nucleic acids (28%), carbohydrates (9%) and lipids (6%). All four types of macromolecules perform 1 or + functions that are not necessary exclusive. However, only the nucleic acids specialize in information storage and transmission. Macromolecules’ functions are directly related to their 3D shapes and to the sequences and chemical properties of their monomers. Polymers are formed by condensation/dehydration reactions. During such reactions, every time a covalent bond is formed between monomers, water is released and energy is added to the system. Polymers are broken down by hydrolysis reactions. During such reactions, water is added to the system and reacts with the covalent bonds that link the polymer together. Hydrolysis releases energy. 3.2. What Are the Chemical Structures and Functions of Proteins Proteins have very diverse functions –only energy storage and information storage do they not perform. They can act as catalysts (enzymes) in biochemical reactions, have a regulation role (e.g. insulin), assure transport of substances (e.g. hemoglobin) and serve many other functions. Proteins are polymers of 20 ≠ AA and consist of 1 or + polypeptide chains. Each chain folds into a specific 3D-shape determined by its sequence of AA. A polypeptide chain can be thought of as a sentence: the “capital letter” is the N-terminus, and the “period” is the C-terminus. Amino acids have carboxyl (-COOH) and amino (-NH ) groups2and thus function as both acid and base. They are typically in ionized forms in living cells and have a positive end (e.g. + - NH )3and a negative end (e.g. COO ). They can exist as optical isomers (D-amino acids – dextro:right –and L-amino acids which are commonly found in the protein of most organisms–levo:left). The side chains (or R-groups) of amino acids also have functional groups that are important in determining the 3D structure of the protein and thus its function. Amino acids can be grouped based on their side chains:  Five amino acids (Arginine, Lysine, Histidine (+), aspartic acid and glutamic acid (-)) have ionized side chains at pH close to 7. They are hydrophilic and attract oppositely charged ions.  Five amino acids (Asparagine, Glutamine, Serine, Threonine and Tyrosine) have polar but uncharged side chains. They are also hydrophilic (because of the high electronegativities of Oxygen and Nitrogen) and attract other polar or charged molecules.  Seven amino acids (including Alanine and Proline) have nonpolar hydrophobic side chains.  Special cases: cysteine (possible disulfide bridge |b| 2 cysteine molecules), glycine (small and fits into tight corners in the interior of proteins), and proline (forms a ring which limits its hydrogen-bonding ability) Amino acids bond together covalently in a condensation reaction during which the carboxyl and amino groups react and a peptide linkage/peptide bond (-CO–NH-) is formed. The primary structure of a protein is its specific sequence of amino acids, which influences both its secondary and tertiary structures –how the protein can twist and fold. Peptide linkages (-N–αC–C-) form the backbone of a protein. A protein’s secondary structure consists of regular, repeated spatial patterns in different regions of a polypeptide chain, and requires hydrogen bonding |b| the amino acids.  α-helix: right-handed coil resulting from hydrogen bonding |b| -NH gps on one amino acid and C=O gps on another  β pleated sheet: formed by 2 or + polypeptide chains that are almost completely extended and aligned; hydrogen bonding |b| the –NH gps on one chain and the C=O gps on the other The four structures of a protein. The specific bending and folding of a protein, giving it its specific and unique shape, constitute its tertiary structure. It often includes a buried interior as well as a surface that is exposed to the environment. This surface presents functional gps that can interact with other molecules in the cell. The tertiary structure is determined and maintained by covalent disulfide bridges, hydrogen bonds |b| side chains, aggregation of hydrophobic side chains in the interior of the protein, and ionic attractions |b| oppositely charged side chains. When a protein is heated, it is said to be denatured: its secondary and tertiary structures break down. When cooled, under normal cellular conditions, the protein’s normal tertiary structure is returned and the protein is functional again. This demonstrates that primary structure specifies tertiary structure: the information for protein folding is contained in the primary structure. A protein’s quaternary structure results from the interaction of its subunits –its polypeptide chains, each of them folded into its own tertiary structure –by hydrophobic interactions, van der Waals forces, ionic attractions and hydrogen bonds. 3.3. What Are the Chemical Structures and Functions of Carbohydrates Carbohydrates: Organic compound containing Carbon, Hydrogen, and Oxygen in the ratio 1:2:1 (general formula C m O 2n nhey have 3 major biochemical roles:  source of stored energy  transport of stored energy  carbon skeletons that can be rearranged to form new molecules. Monosaccharides: simple sugars (e.g. glucose); building blocks of larger carbohydrates. Disaccharides: consist of two monosaccharides linked by covalent bonds (e.g. sucrose). Oligosaccharides: made of 3-20 monosaccharides. Polysaccharides: made up of + than 20 monosaccharides, generally hundreds or thousands (e.g. starch, glycogen, and cellulose). All living cells use glucose (monosaccharide familiarly known as the “blood sugar”) as a source of energy. It exists in a straight-chain form and in two ring forms which are more common (because + stable): α-glucose(aldehyde group below) and β-glucose (aldehyde group above). Monosaccharides are simple sugars made up of varying numbers of carbons: pentoses (five carbons) and hexoses (six carbons), some of which are structural isomers (e.g. Fructose, α- Mannose, and α-Galactose). They are covalently bonded in condensation reactions by glycosidic linkages (either α or β). Oligosaccharides may have additional functional gps. They are often covalently bonded to proteins and lipids on cell surfaces and act as recognition signals. Human blood groups get their specificities from oligosaccharide chains. In contrast to proteins, polysaccharides aren’t always linear chains of monomers; they can be branched. Starch (branched) is the principal energy storage compound for plants as glycogen (highly branched) is for animals. Starch and glycogen are hydrophilic but water-insoluble. However, both polymers are readily hydrolyzed into glucose monomers, making them useful for energy storage. Cellulose is the predominant component of plant cell walls and the most abundant organic compound on Earth. It has a linear macromolecular structure in contrast with starch and glycogen. It is a very stable glucose polysaccharide that can withstand harsh environmental conditions without substantial change. Carbohydrates can be modified by oxidation-reduction reactions, or by the addition of functional groups. - Adding a phosphate gp to 1 or + of the –OH sites results in sugar phosphates. Ex: - Substituting an –OH gp with an amino gp (-NH ) re2ults in amino sugars. Ex: Glucosamine - The polymer chitin that makes up the external skeletons of insects and many crustaceans (and is a component of the cell walls of fungi) presents a derivative of glucosamine. Chitin rivals cellulose as one of the most abundant substances on Earth. 3.4. What Are the Chemical Structures and Functions of Lipids N.B: Lipids are not, in the strict sense of the term, polymers! Lipids (or fats): non polar hydrocarbons, hydrophobic, insoluble in water. When sufficiently close together, weak but additive van der Waals forces help hold them together. They play diverse functions in living organisms:  Fats and oils store energy.  Phospholipids play important structural roles in cell membranes.  Carotenoids and chlorophylls help plants capture light energy.  Steroids and modified fatty acids play regulatory roles as hormones and vitamins.  Animal fat serves as thermal insulation.  A lipid coating around nerves provides electrical insulation.  Oil or wax on skin, fur, and feathers repel water. Fats and oils are triglycerides. - Fats – solid at room temperature ( ≈ 20°C) - Oils – liquid at room temperature Glycerol: alcohol with three hydroxyl groups. Fatty acid: long nonpolar hydrocarbon chain with a polar hydroxyl group. Ester linkage During the condensation reaction, the carboxyl group of a fatty acid bonds with the hydroxyl group of glycerol, resulting in an ester linkage. Fatty acids are amphipathic: they have opposing chemical properties. The long hydrocarbon tail is hydrophobic while the carboxyl group, when ionized, is strongly hydrophilic.  Saturated fatty acids: saturated with hydrogen atoms; no double bonds between carbons; inclined to close packing and forming a solid.  Unsaturated fatty acids: 1 or + double bonds between carbons; have kinks at the end of the chain that prevent close packing. Phospholipids: very much like triglycerides, except with a phosphate group replacing one fatty acid. Their head (the phosphate gp) is hydrophilic while their tail is hydrophobic. Like fatty acids, they are amphipathic. In water, hydrophobic interactions bring phospholipids’ tails together inward, away from water, while their hydrophilic heads face outward and interact with water. They form what is called a phospholipid bilayer. Carotenoids and steroids are other nonpolar and amphipathic lipids that have different structures and roles. Carotenoids: light-absorbing pigments found in plants and animals. In humans, a molecule of β-carotene can be broken down into two vitamin A molecules, required for the synthesis of a pigment responsible for vision. Β-Carotene Chapter 4 4.1 What Are the Chemical Structures and Functions of Nucleic Acids DNA: deoxyribonucleic acid RNA: ribonucleic acid Nucleic Pentose sugar Presence of Bases Strands Acid hydroxyl group on 2’ carbon RNA Ribose Yes Adenine – Single Uracil Cytosine - Guanine DNA Deoxyribose No Adenine – Double Thymine Cytosine – Guanine Table 1: Distinguishing RNA from DNA Nucleotide monomers are the building blocks of DNA and RNA polymers. They consist of a nitrogen-containing base, a pentose sugar, and a phosphate group. Nucleosides differ from nucleotides in that they don’t have a phosphate group. The bases of the nucleic acids take one of two forms: - Pyrimidines (a single-ring structure - the smaller ones): Cytosine, Thymine, and Uracil. - Purines (a fused double-ring structure – the bigger ones): Adenine and Guanine Nucleotides are linked together (in condensation reactions) by phosphodiester linkages. The phosphate on the new nucleotide being added to the existing chain, attached to the 5’ carbon of its sugar, is linked to the 3’-carbon on the last sugar of the chain. The direction of growth of nucleic acids is 5’-3’. Oligonucleotides: relatively short, about 20 nucleotide monomers (e.g. RNA primers). Polynucleotides or nucleic acids: can be very long –up to millions of monomers. Complementary base pairing: purines (A and G) pair with pyrimidines (T, U, and C) by hydrogen bonds. Adenine and Thymine are linked by two bonds, whereas Cytosine and Guanine are linked by three bonds. Although RNA is single-stranded, it can fold back on itself and form double-stranded segments when base pairing occurs. This is responsible for the 3D structure of some RNA molecules. The two strands of DNA are connected into a double helix. All DNA molecules have the same structure. The diversity lies in the nucleotide base sequences. DNA is much longer than RNA and has a greater lifespan as well. DNA is a purely informational molecule: the information is encoded in the sequence of bases. The Central Dogma DNA replication and transcription depend on base pairing. The process of translation does not have its reverse: we can’t go from polypeptides to nucleic acids. N.B: Only relatively small segments of DNA are transcribed into RNA. DNA being inherited, determining the sequence of bases and doing comparisons with them helps reveal evolutionary relationships. That is how we know the closest living relative of humans is the chimpanzee. Nucleotides are not merely the building blocks of nucleic acids. They have other important roles:  ATP – energy transducer in biochemical reactions  GTP – energy source in protein synthesis  cAMP – essential in the actions of hormones and in the transmission of information by the nervous system.

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Chapter 3, Problem 3.5.54 is Solved
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Textbook: Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign)
Edition: 3
Author: David Poole
ISBN: 9780538735452

The answer to “compute the rank and nullity of the given matrices over the indicated p.” is broken down into a number of easy to follow steps, and 13 words. The full step-by-step solution to problem: 3.5.54 from chapter: 3 was answered by , our top Math solution expert on 01/29/18, 04:03PM. This full solution covers the following key subjects: . This expansive textbook survival guide covers 7 chapters, and 1985 solutions. Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) was written by and is associated to the ISBN: 9780538735452. This textbook survival guide was created for the textbook: Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign), edition: 3. Since the solution to 3.5.54 from 3 chapter was answered, more than 309 students have viewed the full step-by-step answer.

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compute the rank and nullity of the given matrices over the indicated p