This section discusses linear combinations and independence of vectors. It explains that determining if vectors are linearly dependent or independent involves solving a linear system of equations. The document then provides examples of checking if sets of vectors are linearly dependent or independent by setting up and row reducing the associated coefficient matrix. It demonstrates that the reduced row echelon form reveals whether a nontrivial solution exists, indicating dependence or independence.
This section introduces linear systems of equations. It provides examples of solving systems by substitution that have unique solutions, no solutions, or infinitely many solutions. These examples are presented as worked problems showing the step-by-step work. The key outcomes are that systems can have a single solution if the lines/planes intersect at a single point, no solution if they are parallel and don't intersect, or infinitely many solutions if they are dependent and coincide. Graphical interpretations of the solutions are discussed for systems of two equations in two variables and systems of three equations in three variables.
This document discusses equations and their solutions. It defines key terms like constants, variables, and types of equations like linear equations with one or two variables. It describes methods for solving different types of equations, including using properties of operations and graphical methods to solve systems of linear equations. Sample problems are worked through applying these methods. The document provides context on the importance of equations throughout history and in various sciences.
This document provides an overview of different methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula. It includes examples of solving quadratic equations using each method. Factorisation involves finding two binomial factors whose product is the quadratic expression. Completing the square transforms the equation into a perfect square plus an extra term, allowing it to be factorised. The quadratic formula provides the general solution for any quadratic equation in the form ax^2 + bx + c = 0.
The document discusses solving polynomial equations. It begins by explaining quadratic equations, including how to solve them by factoring or using the quadratic formula. It then introduces polynomial equations of higher degree and methods for determining their real or complex roots, including the fundamental theorem of algebra. Examples are provided to illustrate solving quadratic and polynomial equations using these various methods.
Folleto de matematicas, primero bgu, proyecto ebja. ing luis panimbozaluisdin2729
The document discusses systems of equations and methods for solving systems of two linear equations with two unknowns. It describes three methods - equalization, addition/subtraction (elimination), and substitution. It provides examples of using each method to solve systems of equations and checks the solutions. Determinants and their use in solving systems via Cramer's Rule are also introduced.
The document discusses solving polynomial and rational inequalities. It provides examples of polynomial inequalities and explains the method of critical points to determine the solution set of a polynomial inequality. The method involves factorizing the associated polynomial equation to find critical points, placing them on the number line based on their multiplicity, and determining the solution set as the intervals with the appropriate sign.
This document summarizes key points about linear differential equations:
1. Students should check that theorems in this section about general solutions of linear equations reduce to those in the previous section for n=2.
2. Examples show finding linear combinations of solutions "by inspection" or trial and error to satisfy given equations.
3. Imposing initial conditions on general solutions yields particular solutions for various differential equations.
This document discusses graphing and solving quadratic inequalities. It provides examples of graphing quadratic inequalities in two variables by drawing the parabola defined by the equation and shading the appropriate region based on the inequality symbol. It also discusses graphing systems of quadratic inequalities by identifying the common region where the individual graphs overlap. The document further explains how to solve quadratic inequalities in one variable either graphically by identifying the x-values where the parabola lies above or below the x-axis, or algebraically by finding the critical values and testing intervals. Examples are provided to illustrate both graphical and algebraic approaches.
This section introduces linear systems of equations. It provides examples of solving systems by substitution that have unique solutions, no solutions, or infinitely many solutions. These examples are presented as worked problems showing the step-by-step work. The key outcomes are that systems can have a single solution if the lines/planes intersect at a single point, no solution if they are parallel and don't intersect, or infinitely many solutions if they are dependent and coincide. Graphical interpretations of the solutions are discussed for systems of two equations in two variables and systems of three equations in three variables.
This document discusses equations and their solutions. It defines key terms like constants, variables, and types of equations like linear equations with one or two variables. It describes methods for solving different types of equations, including using properties of operations and graphical methods to solve systems of linear equations. Sample problems are worked through applying these methods. The document provides context on the importance of equations throughout history and in various sciences.
This document provides an overview of different methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula. It includes examples of solving quadratic equations using each method. Factorisation involves finding two binomial factors whose product is the quadratic expression. Completing the square transforms the equation into a perfect square plus an extra term, allowing it to be factorised. The quadratic formula provides the general solution for any quadratic equation in the form ax^2 + bx + c = 0.
The document discusses solving polynomial equations. It begins by explaining quadratic equations, including how to solve them by factoring or using the quadratic formula. It then introduces polynomial equations of higher degree and methods for determining their real or complex roots, including the fundamental theorem of algebra. Examples are provided to illustrate solving quadratic and polynomial equations using these various methods.
Folleto de matematicas, primero bgu, proyecto ebja. ing luis panimbozaluisdin2729
The document discusses systems of equations and methods for solving systems of two linear equations with two unknowns. It describes three methods - equalization, addition/subtraction (elimination), and substitution. It provides examples of using each method to solve systems of equations and checks the solutions. Determinants and their use in solving systems via Cramer's Rule are also introduced.
The document discusses solving polynomial and rational inequalities. It provides examples of polynomial inequalities and explains the method of critical points to determine the solution set of a polynomial inequality. The method involves factorizing the associated polynomial equation to find critical points, placing them on the number line based on their multiplicity, and determining the solution set as the intervals with the appropriate sign.
This document summarizes key points about linear differential equations:
1. Students should check that theorems in this section about general solutions of linear equations reduce to those in the previous section for n=2.
2. Examples show finding linear combinations of solutions "by inspection" or trial and error to satisfy given equations.
3. Imposing initial conditions on general solutions yields particular solutions for various differential equations.
This document discusses graphing and solving quadratic inequalities. It provides examples of graphing quadratic inequalities in two variables by drawing the parabola defined by the equation and shading the appropriate region based on the inequality symbol. It also discusses graphing systems of quadratic inequalities by identifying the common region where the individual graphs overlap. The document further explains how to solve quadratic inequalities in one variable either graphically by identifying the x-values where the parabola lies above or below the x-axis, or algebraically by finding the critical values and testing intervals. Examples are provided to illustrate both graphical and algebraic approaches.
The document discusses linear pairs of equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0. It explains that a pair of linear equations can be solved either algebraically or graphically. The graphical method involves plotting the lines defined by each equation on a graph and analyzing their intersection. Parallel lines mean no solution, intersecting lines mean a unique solution, and coincident lines mean infinitely many solutions. Several examples are worked through to demonstrate these concepts.
This document introduces concepts related to second-order linear differential equations including superposition of solutions, existence and uniqueness of solutions, linear independence, the Wronskian, and general solutions. It provides 16 examples of imposing initial conditions on general solutions to obtain particular solutions. It also includes problems assessing understanding of related concepts and solving characteristic equations.
The document discusses different types of equations and methods for solving them. It begins by defining key terms like constants, variables, and different types of equations like linear equations with one and two unknowns. It then explains techniques for solving each type, such as using the principle of opposite operations or graphical methods of plotting the equations on a coordinate plane to find the point of intersection. Specific examples are provided to illustrate each problem-solving technique. The document serves to provide an overview of foundational algebraic concepts and methods.
The document introduces concepts related to vector spaces including vectors, linear independence, and subspaces. It provides examples in R3 involving determining if sets of vectors are linearly dependent or independent, finding representations of vectors as linear combinations of other vectors, and solving homogeneous and nonhomogeneous systems of equations involving vector coefficients. Key concepts are illustrated through a series of problems involving vectors in R3.
The document discusses subspaces of vector spaces. It provides examples of subsets of Rn and determines whether each subset is a subspace by checking if it is closed under vector addition and scalar multiplication. Some subsets are shown to be subspaces, while others are not subspaces because they fail to satisfy one of the closure properties. The document also uses row reduction to determine the solution spaces of homogeneous linear systems, which must always be subspaces.
Conceptual Short Tricks for JEE(Main and Advanced)Pony Joglekar
The document contains solutions to multiple trigonometry identity and concept questions. For each question, the solution uses substitution techniques to simplify the expressions and arrive at the answer. Key steps include:
1) Letting variable angles equal specific values like 0, 30, 45, 60, 90 degrees to simplify trig functions.
2) Applying identities like sin^2 x + cos^2 x = 1 to isolate variables.
3) Substituting the simplified expressions back into the original to arrive at an identity equaling the answer choices.
The techniques shown provide concise solutions through strategic substitution of angle values and use of trig identities.
The document provides an overview of solving word problems, explaining the process as reading the problem, representing unknowns with variables, relating the unknowns to given values, writing an equation, solving the equation, and proving the answer. It also defines odd, even, and consecutive numbers and provides examples of representing and solving word problems involving these types of numbers.
This document discusses different methods for solving systems of linear equations: substitution, elimination, graphing, and determinants. It provides examples of each method and the step-by-step processes. The substitution method involves solving one equation for a variable and substituting it into the other equation. The elimination method uses multiplication to eliminate a variable. The determinant method uses determinants of coefficients. The graphing method plots the equations as lines and finds their point of intersection.
The document discusses solving systems of linear equations with two or three variables. There are three possible cases for the solution: 1) a unique solution, 2) infinitely many solutions (a dependent system), or 3) no solution. The document demonstrates solving systems using substitution and elimination methods, and provides examples of each case. Graphically, case 1 corresponds to intersecting lines or planes, case 2 to coinciding lines or intersecting planes, and case 3 to parallel lines or non-intersecting planes.
This document contains an unsolved mathematics paper from 2004 containing 37 multiple choice problems testing critical reasoning skills. Some example problems include finding the digit sum of an arithmetic expression, determining the angle of intersection of two curves, and finding the value of x that satisfies a complex logarithmic equation. The problems cover a wide range of mathematics topics including algebra, trigonometry, logarithms, and geometry.
La siguiente presentación ejecutada por mi persona Angeli Dannielys Peña Suárez, estudiante de la Universidad Politécnica Territorial Andes Eloy Blanco te sera de gran ayuda para saber un poco mas acerca de de los conceptos y ejemplos de los conjuntos, pertenencia, agrupación, intersección, operaciones con conjuntos, los números reales y sus conjuntos, desigualdades, valor absoluto, desigualdades con valor absoluto, plano numérico y las cónicas.
1. The angles labeled (2a)° and (5a + 5)° are supplementary and add up to 180°. Solving for a gives a = 25. Similarly, the angles labeled (4b +10)° and (2b – 10)° are supplementary and add up to 180°. Solving for b gives b = 30. Therefore, a + b = 25 + 30 = 55.
2. Two parallel lines intersected by a transversal form eight angles. The acute angles are equal and the obtuse angles are equal. The acute angles are supplementary to the obtuse angles. Solving the equation relating the angles gives the answer.
3. The relationship between angles formed when
This document provides 30 mathematics questions with multiple choice answers for a JEE Main exam practice test. It includes instructions that there are 120 total marks, each question is worth 4 marks, and a 1/4 mark deduction for incorrect answers. The questions cover a range of mathematics topics including trigonometry, coordinate geometry, algebra, calculus, and probability.
This document defines and explains sets and operations on sets. It begins by defining a set as a collection of objects or elements. It then discusses set notation, listing elements within curly brackets. Various types of sets are defined, including finite, infinite, universal, and empty sets. Methods for defining sets by enumeration or description are presented. Common set operations like union, intersection, difference, and complement are defined using examples and Venn diagrams. Properties of sets and laws of sets such as commutativity, associativity, and distribution are stated. The document also discusses the real number system and subsets of real numbers. It defines absolute value and absolute value inequalities, explaining how to solve such inequalities by considering two cases.
This document contains 3 sections with multiple choice questions about lines and slopes. Section 1 asks about identifying parallel and perpendicular lines based on their equations. Section 2 asks similar questions about parallel lines and finding slopes. Section 3 provides graphical representations of lines and asks about determining properties like slope from the graphs. The questions cover topics like identifying parallel and perpendicular lines, finding missing components of a line's equation, and determining properties of lines from graphs.
i) The document discusses various methods for solving systems of linear equations, including graphing, substitution, elimination, and cross-multiplication.
ii) It also addresses solving systems that can be reduced to linear equations, such as transforming non-linear equations using substitution.
iii) Examples are provided to illustrate each method for deriving the solution of a system of equations.
The document provides an introduction to the binomial theorem. It begins by discussing binomial coefficients through the Pascal's triangle. It then derives an explicit formula for binomial coefficients using factorials. Finally, it states the binomial theorem and provides examples of using it to expand algebraic expressions and estimate numerical values.
This document provides information about a mathematics test from Joglekar Mathematics Point in Kota, India. It includes 30 questions for the JEE Main exam with instructions. Each question is worth 4 marks for a correct response and there is a 1 mark deduction for an incorrect response. The maximum total marks for the test are 120. The document then lists the 30 questions and possible multiple choice answers for each.
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a single linear equation represents a straight line and can have infinitely many solution pairs (x,y). The document also discusses how two linear equations can have a unique solution if their lines intersect, no solution if the lines are parallel, or infinitely many solutions if the lines are coincident. Finally, it presents different algebraic methods for solving systems of two linear equations, including substitution, elimination of coefficients, and cross-multiplication.
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a linear equation can have infinitely many solutions (x,y value pairs) that satisfy the equation, and these solutions lie on a straight line. The document provides an example of a single linear equation and shows its graph on the Cartesian plane. It also discusses systems of two linear equations, explaining that their solutions occur where the lines intersect. The document covers various algebraic methods for solving systems of linear equations, including elimination by substitution or equating coefficients, and solving by cross multiplication. It provides examples to illustrate these solution
The document discusses linear pairs of equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0. It explains that a pair of linear equations can be solved either algebraically or graphically. The graphical method involves plotting the lines defined by each equation on a graph and analyzing their intersection. Parallel lines mean no solution, intersecting lines mean a unique solution, and coincident lines mean infinitely many solutions. Several examples are worked through to demonstrate these concepts.
This document introduces concepts related to second-order linear differential equations including superposition of solutions, existence and uniqueness of solutions, linear independence, the Wronskian, and general solutions. It provides 16 examples of imposing initial conditions on general solutions to obtain particular solutions. It also includes problems assessing understanding of related concepts and solving characteristic equations.
The document discusses different types of equations and methods for solving them. It begins by defining key terms like constants, variables, and different types of equations like linear equations with one and two unknowns. It then explains techniques for solving each type, such as using the principle of opposite operations or graphical methods of plotting the equations on a coordinate plane to find the point of intersection. Specific examples are provided to illustrate each problem-solving technique. The document serves to provide an overview of foundational algebraic concepts and methods.
The document introduces concepts related to vector spaces including vectors, linear independence, and subspaces. It provides examples in R3 involving determining if sets of vectors are linearly dependent or independent, finding representations of vectors as linear combinations of other vectors, and solving homogeneous and nonhomogeneous systems of equations involving vector coefficients. Key concepts are illustrated through a series of problems involving vectors in R3.
The document discusses subspaces of vector spaces. It provides examples of subsets of Rn and determines whether each subset is a subspace by checking if it is closed under vector addition and scalar multiplication. Some subsets are shown to be subspaces, while others are not subspaces because they fail to satisfy one of the closure properties. The document also uses row reduction to determine the solution spaces of homogeneous linear systems, which must always be subspaces.
Conceptual Short Tricks for JEE(Main and Advanced)Pony Joglekar
The document contains solutions to multiple trigonometry identity and concept questions. For each question, the solution uses substitution techniques to simplify the expressions and arrive at the answer. Key steps include:
1) Letting variable angles equal specific values like 0, 30, 45, 60, 90 degrees to simplify trig functions.
2) Applying identities like sin^2 x + cos^2 x = 1 to isolate variables.
3) Substituting the simplified expressions back into the original to arrive at an identity equaling the answer choices.
The techniques shown provide concise solutions through strategic substitution of angle values and use of trig identities.
The document provides an overview of solving word problems, explaining the process as reading the problem, representing unknowns with variables, relating the unknowns to given values, writing an equation, solving the equation, and proving the answer. It also defines odd, even, and consecutive numbers and provides examples of representing and solving word problems involving these types of numbers.
This document discusses different methods for solving systems of linear equations: substitution, elimination, graphing, and determinants. It provides examples of each method and the step-by-step processes. The substitution method involves solving one equation for a variable and substituting it into the other equation. The elimination method uses multiplication to eliminate a variable. The determinant method uses determinants of coefficients. The graphing method plots the equations as lines and finds their point of intersection.
The document discusses solving systems of linear equations with two or three variables. There are three possible cases for the solution: 1) a unique solution, 2) infinitely many solutions (a dependent system), or 3) no solution. The document demonstrates solving systems using substitution and elimination methods, and provides examples of each case. Graphically, case 1 corresponds to intersecting lines or planes, case 2 to coinciding lines or intersecting planes, and case 3 to parallel lines or non-intersecting planes.
This document contains an unsolved mathematics paper from 2004 containing 37 multiple choice problems testing critical reasoning skills. Some example problems include finding the digit sum of an arithmetic expression, determining the angle of intersection of two curves, and finding the value of x that satisfies a complex logarithmic equation. The problems cover a wide range of mathematics topics including algebra, trigonometry, logarithms, and geometry.
La siguiente presentación ejecutada por mi persona Angeli Dannielys Peña Suárez, estudiante de la Universidad Politécnica Territorial Andes Eloy Blanco te sera de gran ayuda para saber un poco mas acerca de de los conceptos y ejemplos de los conjuntos, pertenencia, agrupación, intersección, operaciones con conjuntos, los números reales y sus conjuntos, desigualdades, valor absoluto, desigualdades con valor absoluto, plano numérico y las cónicas.
1. The angles labeled (2a)° and (5a + 5)° are supplementary and add up to 180°. Solving for a gives a = 25. Similarly, the angles labeled (4b +10)° and (2b – 10)° are supplementary and add up to 180°. Solving for b gives b = 30. Therefore, a + b = 25 + 30 = 55.
2. Two parallel lines intersected by a transversal form eight angles. The acute angles are equal and the obtuse angles are equal. The acute angles are supplementary to the obtuse angles. Solving the equation relating the angles gives the answer.
3. The relationship between angles formed when
This document provides 30 mathematics questions with multiple choice answers for a JEE Main exam practice test. It includes instructions that there are 120 total marks, each question is worth 4 marks, and a 1/4 mark deduction for incorrect answers. The questions cover a range of mathematics topics including trigonometry, coordinate geometry, algebra, calculus, and probability.
This document defines and explains sets and operations on sets. It begins by defining a set as a collection of objects or elements. It then discusses set notation, listing elements within curly brackets. Various types of sets are defined, including finite, infinite, universal, and empty sets. Methods for defining sets by enumeration or description are presented. Common set operations like union, intersection, difference, and complement are defined using examples and Venn diagrams. Properties of sets and laws of sets such as commutativity, associativity, and distribution are stated. The document also discusses the real number system and subsets of real numbers. It defines absolute value and absolute value inequalities, explaining how to solve such inequalities by considering two cases.
This document contains 3 sections with multiple choice questions about lines and slopes. Section 1 asks about identifying parallel and perpendicular lines based on their equations. Section 2 asks similar questions about parallel lines and finding slopes. Section 3 provides graphical representations of lines and asks about determining properties like slope from the graphs. The questions cover topics like identifying parallel and perpendicular lines, finding missing components of a line's equation, and determining properties of lines from graphs.
i) The document discusses various methods for solving systems of linear equations, including graphing, substitution, elimination, and cross-multiplication.
ii) It also addresses solving systems that can be reduced to linear equations, such as transforming non-linear equations using substitution.
iii) Examples are provided to illustrate each method for deriving the solution of a system of equations.
The document provides an introduction to the binomial theorem. It begins by discussing binomial coefficients through the Pascal's triangle. It then derives an explicit formula for binomial coefficients using factorials. Finally, it states the binomial theorem and provides examples of using it to expand algebraic expressions and estimate numerical values.
This document provides information about a mathematics test from Joglekar Mathematics Point in Kota, India. It includes 30 questions for the JEE Main exam with instructions. Each question is worth 4 marks for a correct response and there is a 1 mark deduction for an incorrect response. The maximum total marks for the test are 120. The document then lists the 30 questions and possible multiple choice answers for each.
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a single linear equation represents a straight line and can have infinitely many solution pairs (x,y). The document also discusses how two linear equations can have a unique solution if their lines intersect, no solution if the lines are parallel, or infinitely many solutions if the lines are coincident. Finally, it presents different algebraic methods for solving systems of two linear equations, including substitution, elimination of coefficients, and cross-multiplication.
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a linear equation can have infinitely many solutions (x,y value pairs) that satisfy the equation, and these solutions lie on a straight line. The document provides an example of a single linear equation and shows its graph on the Cartesian plane. It also discusses systems of two linear equations, explaining that their solutions occur where the lines intersect. The document covers various algebraic methods for solving systems of linear equations, including elimination by substitution or equating coefficients, and solving by cross multiplication. It provides examples to illustrate these solution
1) The document provides information about linear equations in two variables including the general form of a linear equation, single linear equations, systems of two linear equations, conditions for common solutions, and methods to solve systems of linear equations algebraically.
2) Examples are provided to illustrate graphing single linear equations, finding common solutions to systems of two linear equations, and solving systems using elimination and cross-multiplication methods.
3) Key methods for solving systems of linear equations discussed include elimination by substitution or equating coefficients, and cross-multiplication. Conditions for common solutions depend on whether lines intersect, are parallel, or are coincident.
The Wronskian of two functions y1 and y2, denoted W(y1,y2), determines whether y1 and y2 are linearly independent solutions to a differential equation. If W(y1,y2) is nonzero at some point in the interval, then y1 and y2 are linearly independent over that entire interval. Additionally, y1 and y2 are linearly dependent over an interval if and only if W(y1,y2) is zero for all points in that interval. The Wronskian also determines whether y1 and y2 form a fundamental set of solutions.
This document discusses different methods for solving simultaneous equations and quadratic equations, including:
- Solving simultaneous equations with three unknowns using matrices, determinants, or Cramer's rule. Examples are provided.
- Solving quadratic equations by factorizing (if possible), completing the square, using the quadratic formula, or graphically. Factorization is introduced as the simplest method when applicable.
- Key steps are outlined for each method, such as writing the equations in standard form and determining relevant determinants or matrices. Applications to circuit analysis and mechanical systems are mentioned.
This document provides solutions to problems from Gilbert Strang's Linear Algebra textbook. It derives the decomposition of a matrix A into its basis for the row space and nullspace. It then provides solutions to several problems from Chapter 1 on vectors and Chapter 2 on solving linear equations. The problems cover topics like counting dimensions, vector sums representing hours in a day, and checking properties of subspaces. The document uses notation like ⇒ to represent row reduction steps and derives expressions for matrix inverses and solutions to systems of equations.
Power Point Presentation on a PAIR OF LINEAR EQUATION IN TWO VARIABLES, MATHS project...
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This document contains solutions to selected exercises from the textbook "Principles of Digital Communication" by Robert G. Gallager. The solutions cover material from Chapter 2, including properties of random variables, expectation, independence, coding theorems, Huffman coding, and entropy bounds. Specific solutions include showing that the variance of the sum of independent random variables is the sum of the variances, providing an example where expectation is not multiplicative, and proving properties of prefix-free and suffix-free codes.
1. The question provides information about a past UPSEE mathematics exam from 2006 containing 40 multiple choice questions covering topics in complex numbers, trigonometry, geometry, and coordinate geometry.
2. For each question, four possible answers (a, b, c, or d) are provided and test-takers must indicate the correct answer in their answer book.
3. The document contains 40 multiple choice questions testing a range of mathematical concepts and skills.
This document provides examples and explanations for evaluating double and triple integrals using Cartesian and polar coordinates. It begins by introducing double and triple integrals and their notation. It then discusses the evaluation of double and triple integrals, including the process of integrating inner integrals first and noting that integral limits should proceed from variable to constant. Several examples are worked through to demonstrate evaluating double and triple integrals over different regions of integration. The document also covers changing the order of integration and evaluating area integrals using double integrals in both Cartesian and polar coordinate systems.
The document discusses linear equations in two variables. It defines linear equations as equations containing two variables where each variable has an exponent of 1. It provides examples and discusses the general form of simultaneous linear equations as a1x + b1y = c1 and a2x + b2y = c2. The document also discusses framing linear equations from word problems, graphically representing solutions, criteria for consistent/inconsistent systems, and methods for algebraically solving simultaneous linear equations including elimination, substitution, and cross multiplication.
1. The document discusses various types of transformations in complex analysis, including translation, rotation, stretching, and inversion.
2. Under inversion (1/w=z), a straight line is mapped to a circle if it does not pass through the origin, and to another straight line if it does pass through the origin. A circle is always mapped to another circle.
3. A general bilinear or Möbius transformation can be expressed as a combination of translation, rotation, stretching, and inversion.
Advanced Engineering Mathematics Solutions Manual.pdfWhitney Anderson
This document contains 27 multi-part exercises involving differential equations. The exercises cover topics such as determining whether differential equations are linear or nonlinear, solving differential equations, and classifying differential equations by order.
This document provides a summary of lecture 2 on quadratic equations and straight lines. It covers how to factorize, complete the square, and use the quadratic formula to solve quadratic equations. It also discusses how to find the equation of a straight line given its gradient and y-intercept, or two points on the line. Additionally, it explains how to sketch lines, find the midpoint and distance between two points. Key terms defined include quadratic, surd, gradient, and intercept. Methods demonstrated include solving quadratic equations, finding lines from gradient/point and two points, and calculating midpoints and distances on a graph.
The document provides information about module 1 on plane coordinate geometry. It will explain the relationship between lines on a plane, including intersecting, parallel and perpendicular lines. It will also cover determining the point of intersection between two lines algebraically and identifying if lines are parallel, perpendicular or neither based on their equations. Examples are provided to find the intersection of lines and to determine if lines are parallel, perpendicular or intersecting without graphing.
This document is the preface to the instructor's manual for Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion. It provides an overview of the contents of the manual, which contains solutions to the end-of-chapter problems from the textbook. The preface notes there are now 509 problems and the solutions range from straightforward to challenging. It stresses the solutions are only for instructors and should not be shared with students.
Order of presentation
Anushka - Opening
Nikunj -Intro
Shubham - Graphical
Amel - Sunstitution
Siddhartha- Elimination
Karthik - Cross multiplication
Anushka - Equations reducible...& wrap-up
In case of any confusion..inform me by facebook, phone or in school
This document contains 60 multiple choice questions from a past mathematics exam. The questions cover a range of topics including relations, functions, complex numbers, matrices, determinants, quadratic equations, arithmetic and geometric progressions, binomial expansions, trigonometry, calculus, differential equations, vectors, conic sections, and three-dimensional geometry. For each question, four choices are given and the student must select the correct answer.
The document outlines the learning objectives and test questions for a chapter on controlling in organizations. It covers explaining the foundations of control, identifying the phases of the corrective control model, describing primary control methods, and explaining corporate governance issues. The questions are categorized by their learning objective, type (true/false, multiple choice, essay), and difficulty level (easy, moderate, difficult).
This document provides an overview of organizational culture and cultural diversity. It includes learning objectives, test correlation tables, true/false questions, and multiple choice question previews related to:
1) Describing the core elements of organizational culture including symbols, language, values, norms, and narratives.
2) Comparing and contrasting four types of organizational culture: clan, adhocracy, market, and hierarchy.
3) Discussing several types of subcultures that may exist within organizations including departmental, generational, and gender-based subcultures.
4) Describing several activities for successfully managing diversity such as surveys, training, and establishing employee resource groups.
This document contains a chapter on work motivation from a textbook. It includes learning objectives about different theories of motivation, such as the managerial approach, reinforcement theory, expectancy theory, and job characteristics theory. There are also true/false questions and multiple choice questions testing understanding of these motivation theories.
This document provides a test correlation table and learning objectives for a chapter on managing human resources. The table lists the chapter's learning objectives and correlates them with different types of test questions (true/false, multiple choice, essay) at different levels of difficulty (easy, moderate, difficult). It then provides examples of test questions for each objective, including the question, answer, rationale, and difficulty level. The chapter appears to cover topics like the strategic importance of human resources, employment laws and regulations, human resources planning, recruitment and hiring, training and development, performance appraisals, and compensation.
This document provides an overview of managing work teams. It begins with learning objectives about explaining the importance of work teams, identifying types of work teams, stating the meaning and determinants of team effectiveness, describing internal team processes, and explaining how to diagnose and remove barriers to performance. It then provides a correlation table matching questions to these learning objectives at different levels of difficulty. The remainder of the document consists of true/false questions mapping to the learning objectives.
This document contains a chapter on organizational communication from a textbook. It includes 42 true/false questions testing comprehension of key concepts about communication processes in organizations. It also includes 23 multiple choice questions assessing understanding of topics like encoding, decoding, channels, and types of messages used. Key points covered include the importance of communication in organizations, elements of the communication process, and the role of both verbal and nonverbal communication.
This document contains a chapter on leadership dynamics with 6 learning objectives. It provides true/false and multiple choice questions with answers on theories and models of leadership, including:
- Leadership involves influence, change, and shared purpose between leaders and followers.
- Behavioral models show leadership behaviors can be learned and focus on differences between effective and ineffective leaders.
- Contingency models like Situational Leadership state the best leadership style depends on the situation.
- Transformational leaders inspire followers through vision and innovation.
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Sect4 3
1. SECTION 4.3
LINEAR COMBINATIONS AND
INDEPENDENCE OF VECTORS
In this section we use two types of computational problems as aids in understanding linear
independence and dependence. The first of these problems is that of expressing a vector w as a
linear combination of k given vectors v1 , v 2 , , v k (if possible). The second is that of
determining whether k given vectors v1 , v 2 , , v k are linearly independent. For vectors in Rn,
each of these problems reduces to solving a linear system of n equations in k unknowns. Thus an
abstract question of linear independence or dependence becomes a concrete question of whether or
not a given linear system has a nontrivial solution.
1. v2 = 3
2
v1 , so the two vectors v1 and v2 are linearly dependent.
2. Evidently the two vectors v1 and v2 are not scalar multiples of one another. Hence they
are linearly dependent.
3. The three vectors v1, v2, and v3 are linearly dependent, as are any 3 vectors in R2. The
reason is that the vector equation c1v1 + c2v2 + c3v3 = 0 reduces to a homogeneous linear
system of 2 equations in the 3 unknowns c1 , c2 , and c3 , and any such system has a
nontrivial solution.
4. The four vectors v1, v2, v3, and v4 are linearly dependent, as are any 4 vectors in R3. The
reason is that the vector equation c1v1 + c2v2 + c3v3 + c4v4 = 0 reduces to a homogeneous
linear system of 3 equations in the 4 unknowns c1 , c2 , c3 , and c4 , and any such system
has a nontrivial solution.
5. The equation c1 v1 + c2 v 2 + c3 v 3 = 0 yields
c1 (1, 0, 0) + c2 (0, −2, 0) + c3 (0, 0,3) = (c1 , −2c2 ,3c3 ) = (0, 0, 0),
and therefore implies immediately that c1 = c2 = c3 = 0. Hence the given vectors
v1, v2, and v3 are linearly independent.
6. The equation c1 v1 + c2 v 2 + c3 v 3 = 0 yields
c1 (1, 0, 0) + c2 (1,1, 0) + c3 (1,1,1) = (c1 + c2 + c3 , c2 + c3 , c3 ) = (0, 0, 0).
But it is obvious by back-substitution that the homogeneous system
2. c1 + c2 + c3 = 0
c2 + c3 = 0
c3 = 0
has only the trivial solution c1 = c2 = c3 = 0. Hence the given vectors
v1, v2, and v3 are linearly independent.
7. The equation c1 v1 + c2 v 2 + c3 v 3 = 0 yields
c1 (2,1, 0, 0) + c2 (3, 0,1, 0) + c3 (4, 0, 0,1) = (2c1 + 3c2 , c1 , c2 , c3 ) = (0, 0, 0, 0).
Obviously it follows immediately that c1 = c2 = c3 = 0. Hence the given vectors
v1, v2, and v3 are linearly independent.
8. Here inspection of the three given vectors reveals that v 3 = v1 + v 2 , so the vectors
v1, v2, and v3 are linearly dependent.
In Problems 9-16 we first set up the linear system to be solved for the linear combination
coefficients {ci }, and then show the reduction of its augmented coefficient matrix A to reduced
echelon form E.
9. c1 v1 + c2 v 2 = w
5 3 1 1 0 2
A = 3 2 0 → 0 1 −3 = E
4 5 −7
0 0 0
We see that the system of 3 equations in 2 unknowns has the unique solution
c1 = 2, c2 = −3, so w = 2 v1 − 3v 2 .
10. c1 v1 + c2 v 2 = w
−3 6 3 1 0 7
1 −2 −1 → 0 1 4 = E
A =
−2 3 −2
0 0 0
We see that the system of 3 equations in 2 unknowns has the unique solution
c1 = 7, c2 = 4, so w = 7 v1 + 4 v 2 .
3. 11. c1 v1 + c2 v 2 = w
7 3 1 1 0 1
−6 −3 0
A = → 0 1 −2 = E
4 2 0 0 0 0
5 3 −1 0 0 0
We see that the system of 4 equations in 2 unknowns has the unique solution
c1 = 1, c2 = −2, so w = v1 − 2 v 2 .
12. c1 v1 + c2 v 2 = w
7 −2 4 1 0 2
3 −2 −4 5
A = → 0 1 = E
−1 1 3 0 0 0
9 −3 3 0 0 0
We see that the system of 4 equations in 2 unknowns has the unique solution
c1 = 2, c2 = 5, so w = 2 v1 + 5v 2 .
13. c1 v1 + c2 v 2 = w
1 5 5 1 0 0
A = 5 −3 2 → 0 1 0 = E
−3 4 −2
0 0 1
The last row of E corresponds to the scalar equation 0c1 + 0c2 = 1, so the system of 3
equations in 2 unknowns is inconsistent. This means that w cannot be expressed as a
linear combination of v1 and v2.
14. c1v1 + c2 v 2 + c3 v 3 = w
1 0 0 2 1 0 0 0
0 1 − 1 − 3 0
A = → 0 1 0 = E
0 −2 1 2 0 0 1 0
3 0 1 −3 0 0 0 1
The last row of E corresponds to the scalar equation 0c1 + 0c2 + 0c3 = 1, so the system
of 4 equations in 3 unknowns is inconsistent. This means that w cannot be expressed as
a linear combination of v1, v2, and v3.
4. 15. c1v1 + c2 v 2 + c3 v 3 = w
2 3 1 4 1 0 0 3
−1 0 2 5 → 0 1 0 −2 = E
A =
4 1 −1 6
0 0 1 4
We see that the system of 3 equations in 3 unknowns has the unique solution
c1 = 3, c2 = −2, c3 = 4, so w = 3v1 − 2 v 2 + 4 v 3 .
16. c1v1 + c2 v 2 + c3 v 3 = w
2 4 1 7 1 0 0 6
0 1 3 7 0 1 0 −2
A = → = E
3 3 −1 9 0 0 1 3
1 2 3 11 0 0 0 0
We see that the system of 4 equations in 3 unknowns has the unique solution
c1 = 6, c2 = −2, c3 = 3, so w = 6 v1 − 2 v 2 + 3v 3 .
In Problems 17-22, A = [ v1 v2 v 3 ] is the coefficient matrix of the homogeneous linear
system corresponding to the vector equation c1v1 + c2 v 2 + c3 v 3 = 0. Inspection of the indicated
reduced echelon form E of A then reveals whether or not a nontrivial solution exists.
1 2 3 1 0 0
17. 0 −3 5 → 0 1 0 = E
A =
1 4 2
0 0 1
We see that the system of 3 equations in 3 unknowns has the unique solution
c1 = c2 = c3 = 0, so the vectors v1 , v 2 , v 3 are linearly independent.
2 4 −2 1 0 −3 / 5
18. 0 −5 1 → 0 1 −1/ 5 = E
A =
−3 −6 3
0 0
0
We see that the system of 3 equations in 3 unknowns has a 1-dimensional solution space.
If we choose c3 = 5 then c1 = 3 and c2 = 1. Therefore 3v1 + v 2 + 5 v3 = 0.
2 5 2 1 0 0
0 4 −1 0
19. A = → 0 1 = E
3 −2 1 0 0 1
0 1 −1 0 0 0
5. We see that the system of 4 equations in 3 unknowns has the unique solution
c1 = c2 = c3 = 0, so the vectors v1 , v 2 , v 3 are linearly independent.
1 2 3 1 0 0
1 1
1 0 1 0
20. A = → = E
−1 1 4 0 0 1
1 1 1 0 0 0
We see that the system of 4 equations in 3 unknowns has the unique solution
c1 = c2 = c3 = 0, so the vectors v1 , v 2 , v 3 are linearly independent.
3 1 1 1 0 1
0 −1
2 0 1 −2
21. A = → = E
1 0 1 0 0 0
2 1 0 0 0 0
We see that the system of 4 equations in 3 unknowns has a 1-dimensional solution space.
If we choose c3 = −1 then c1 = 1 and c2 = −2. Therefore v1 − 2 v 2 − v 3 = 0.
3 3 5 1 0 7 / 9
9 0
7 0 1 5 / 9
22. A = → = E
0 9 5 0 0 0
5 −7 0 0 0 0
We see that the system of 4 equations in 3 unknowns has a 1-dimensional solution space.
If we choose c3 = −9 then c1 = 7 and c2 = 5. Therefore 7 v1 + 5 v 2 − 9 v3 = 0.
23. Because v1 and v2 are linearly independent, the vector equation
c1u1 + c2u 2 = c1 ( v1 + v 2 ) + c2 ( v1 − v 2 ) = 0
yields the homogeneous linear system
c1 + c2 = 0
c1 − c2 = 0.
It follows readily that c1 = c2 = 0, and therefore that the vectors u1 and u2 are linearly
independent.
24. Because v1 and v2 are linearly independent, the vector equation
c1u1 + c2u 2 = c1 ( v1 + v 2 ) + c2 (2 v1 + 3v 2 ) = 0
6. yields the homogeneous linear system
c1 + 2c2 = 0
c1 + 3c2 = 0.
Subtraction of the first equation from the second one gives c2 = 0, and then it follows
from the first equation that c2 = 0 also. Therefore the vectors u1 and u2 are linearly
independent.
25. Because the vectors v1 , v 2 , v 3 are linearly independent, the vector equation
c1u1 + c2u 2 + c3u 3 = c1 ( v1 ) + c2 ( v1 + 2 v 2 ) + c3 ( v1 + 2 v 2 + 3v 3 ) = 0
yields the homogeneous linear system
c1 + c2 + c3 = 0
2c2 + 2c3 = 0
3c3 = 0.
It follows by back-substitution that c1 = c2 = c3 = 0, and therefore that the vectors
u1 , u 2 , u3 are linearly independent.
26. Because the vectors v1 , v 2 , v 3 are linearly independent, the vector equation
c1u1 + c2u 2 + c3u 3 = c1 ( v 2 + v 3 ) + c2 ( v1 + v 3 ) + c3 ( v1 + v 2 ) = 0
yields the homogeneous linear system
c2 + c3 = 0
c1 + c3 = 0
c1 + c2 = 0.
The reduction
0 1 1 1 0 0
1 0 1 → 0 1 0 = E
A =
1 1 0
0 0 1
then shows that c1 = c2 = c3 = 0, and therefore that the vectors u1 , u 2 , u3 are linearly
independent.
7. 27. If the elements of S are v1 , v 2 , , v k with v1 = 0, then we can take c1 = 1 and
c2 = = ck = 0. This choice gives coefficients c1 , c2 , , ck not all zero such that
c1v1 + c2 v 2 + + ck v k = 0. This means that the vectors v1 , v 2 , , v k are linearly
dependent.
28. Because the set S of vectors v1 , v 2 , , v k is linearly dependent, there exist scalars
c1 , c2 , , ck not all zero such that c1v1 + c2 v 2 + + ck v k = 0. If ck +1 = = cm = 0,
then c1 v1 + c2 v 2 + + cm v m = 0 with the coefficients c1 , c2 , , cm not all zero. This
means that the vectors v1 , v 2 , , v m comprising T are linearly dependent.
29. If some subset of S were linearly dependent, then Problem 28 would imply immediately
that S itself is linearly dependent (contrary to hypothesis).
30. Let W be the subspace of V spanned by the vectors v1 , v 2 , , v k . Because U is a
subspace containing each of these vectors, it contains every linear combination of
v1 , v 2 , , v k . But W consists solely of such linear combinations, so it follows that U
contains W.
31. If S is contained in span(T), then every vector in S is a linear combination of vectors in
T. Hence every vector in span(S) is a linear combination of linear combinations of
vectors in T. Therefore every vector in span(S) is a linear combination of vectors in T,
and therefore is itself in span(T). Thus span(S) is a subset of span(T).
32. If u is another vector in S then the k+1 vectors v1 , v 2 , , v k , u are linearly
dependent. Hence there exist scalars c1 , c2 , , ck , c not all zero such that
c1v1 + c2 v 2 + + ck v k + cu = 0. If c = 0 then we have a contradiction to the
hypothesis that the vectors v1 , v 2 , , v k are linearly independent. Therefore c ≠ 0,
so we can solve for u as a linear combination of the vectors v1 , v 2 , , v k .
33. The determinant of the k × k identity matrix is nonzero, so it follows immediately from
Theorem 3 in this section that the vectors v1 , v 2 , , v k are linearly independent.
34. If the vectors v1 , v 2 , , v n are linearly independent, then by Theorem 2 the matrix
A = [ v1 v 2 v n ] is nonsingular. If B is another nonsingular n × n matrix, then
the product AB is also nonsingular, and therefore (by Theorem 2) has linearly
independent column vectors.
35. Because the vectors v1 , v 2 , , v k are linearly independent, Theorem 3 implies that some
k × k submatrix A0 of A has nonzero determinant. Let A0 consist of the rows
i1 , i2 , , ik of the matrix A, and let C0 denote the k × k submatrix consisting of the
8. same rows of the product matrix C = AB. Then C0 = A0B, so C0 = A 0 B ≠ 0
because (by hypothesis) the k × k matrix B is also nonsingular. Therefore Theorem 3
implies that the column vectors of AB are linearly independent.