Access Elementary Linear Algebra with Applications 9th Edition solutions now. Our solutions are written by Chegg experts so you can be assured of the highest . Instructor’s Solutions Manual (Download only) for Elementary Linear Algebra with Applications, 9th Edition. Bernard Kolman, Drexel University. © | Pearson. Chapter 1 Systems of Linear Equations and Matrices Section Exercise Set 1. (a), (c), and (f) are linear equations in x1, x2, and x3. (b) is not linear.
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In mathematicsan equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: Libear identity is true for all values of the variable.
A conditional equation is true elementqry only particular values of the variables. The expressions on the two sides of the equals sign are called the “left-hand side” and “right-hand side” of the equation. The most common type of equation is an algebraic equationin which the two lagebra are algebraic expressions. Each side of an algebraic equation will contain one or more terms.
For example, the equation. The unknowns are x and y and the parameters are ABand C. An equation is analogous to a scale into which weights are placed. When equal weights of something grain elmeentary example are placed into the two pans, the two weights cause sllutions scale to be in solitions and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. Likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain true.
In geometryequations are used to describe geometric figures. As the equations that are considered, such as implicit equations or parametric equationshave infinitely many solutions, the objective is now different: This is the starting idea of algebraic geometryan important area of mathematics. Algebra studies two main families of equations: To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis. Algebra also studies Diophantine equations where the coefficients and solutions are integers.
The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.
Differential equations are equations that involve one lindar more functions and their derivatives. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics. An equation is analogous to a weighing scale solutioons, balance, solutione seesaw.
Each side of the equation corresponds to one side of the balance. Different quantities can be placed on each side: In the illustration, xy and z are all different quantities in this case real numbers represented as circular weights, and each of xyand z has a different weight. Addition corresponds to adding weight, while subtraction lineqr to removing weight from what is already there.
When equality holds, the total weight on each side is the same. Equations often contain terms other than the unknowns. These other terms, which are assumed to be knownare usually called constantscoefficients or parameters. An example of an equation involving x and y as unknowns and the parameter R is.
Hence, the equation with R unspecified is the general equation for the circle. Usually, the unknowns are denoted by letters at the end of the alphabet, xyzw…, while coefficients parameters are denoted by letters at the beginning, abcd…. The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters is called solving the equation. Such expressions of the solutions in terms of the parameters are also called solutions.
A system of equations is a set of simultaneous equationsusually in several unknowns, for which the common solutions are sought.
Thus a solution to the system is a set of values for each of the unknowns, which together form a solution to each equation in the system. For example, the system. An identity is an equation that is true for all possible values of the variable s it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation making it more easily solvable.
In algebra, an example of an identity is the difference of two squares:. Trigonometry is an area where many identities exist; these are useful in manipulating or solving trigonometric equations.
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Two of many that involve the sine and cosine functions are:. Two equations or two systems of equations are equivalent if they have the same set of solutions. The algebrra operations transform an equation or a system of equations into an equivalent one — provided that the operations are meaningful for the expressions they are applied to:.
If some function is applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but aith have further solutions called extraneous solutions. Thus, caution must be exercised when applying such a transformation to an equation. The above transformations are the basis of most elementary methods for equation solving as well as some less elementary ones, like Gaussian elimination. In general, an algebraic equation or polynomial equation is an equation of the form.
An algebraic equation is univariate if it involves only one variable.
On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate multiple variables, x, y, z, etc. The term polynomial equation is usually preferred to algebraic equation.
Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression with a finite number of operations involving just those coefficients that is, it can be solved algebraically. This can be done applixations all such lineaf of degree one, two, three, or four; but for degree five or more it can be solved for some equations but, as the Abel—Ruffini theorem demonstrates, not for all.
A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation see Root-finding algorithm and of the common solutions of several multivariate polynomial equations see System of polynomial equations.
A system of linear equations or linear system is a collection of linear equations involving the same set of variables. A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by. The word ” system ” indicates that the equations are to be considered collectively, rather than individually.
In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebraa subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebraand play a prominent role in physicsengineeringchemistrycomputer scienceand economics. A system of non-linear equations can often be approximated by a linear system see linearizationa helpful technique when making a mathematical model or computer simulation of a relatively complex system.
In Euclidean geometryit is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid. This method allows one to characterize geometric figures by equations. In other words, in space, all conics are defined as the solution set of an equation of a plane and of the equation of a cone just given. This formalism allows one to determine the positions and the properties of the focuses of a conic.
The use of equations allows one to call on a large area of mathematics to solve geometric questions. The Cartesian coordinate system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the name analytic geometry. This point of view, outlined by Descartesenriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.
Currently, analytic geometry designates an active branch of mathematics. Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as functional analysis and linear algebra.
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinateswhich are the signed distances from the point to two fixed perpendicular directed lines, that are marked using the same unit of length. One can use the same principle to specify the position of any point in three- dimensional space by the use of three Cartesian coordinates, which are the signed distances to three mutually perpendicular planes or, equivalently, by its perpendicular projection onto three mutually perpendicular lines.
Cartesius revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.
Using the Cartesian coordinate system, geometric shapes such as curves can be described by Cartesian equations: A parametric equation for a curve expresses the coordinates of the points of the curve as functions of a variablecalled a parameter. Together, these equations are called a parametric representation of the curve.
The notion of parametric equation has been generalized to surfacesmanifolds and algebraic varieties of higher dimensionwith the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc. A Diophantine equation is a polynomial equation in two or more unknowns for which only the integer solutions are sought an integer solution is a solution such that all the unknowns take integer values.
A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.
Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curvealgebraic surfaceor more witj object, and qpplications about the lattice points on it.
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The word Diophantine refers to algebraa Hellenistic mathematician of the 3rd century, Diophantus of Alexandriawho made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.
An algebraic number is a number that is a solution of a non-zero polynomial equation in one variable with rational coefficients or equivalently — by clearing a;plications — with integer coefficients. Almost all real and complex numbers are transcendental.
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Algebraic geometry is a branch of mathematicsclassically studying solutions of polynomial equations. Modern algebraic geometry is based on more abstract techniques of abstract algebraespecially commutative algebrawith the language and the problems of geometry.
The fundamental objects of study in algebraic geometry are algebraic varietieswhich are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: A point wlth the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation.
Basic questions involve the study of the points of special interest like the singular pointsthe inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.