The document contains announcements and information about an exam for a class. It includes the following key points:
- Students should bring any grade-related questions about Exam 1 without delay. The homework for Exam 2 has been uploaded.
- The professor is planning to cover chapters 3, 5, and 6 for Exam 2.
- The last day for students to drop the class with a grade of "W" is February 4th.
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
- Functions can be represented by ordered pairs, graphs, or equations
- The vertical line test identifies functions as those where a vertical line intersects the graph at most once
- Intercepts occur where the graph crosses the x or y-axis
This document summarizes Chapter 10 from a mathematics textbook. The chapter covers limits and continuity. It introduces limits, such as one-sided limits and limits at infinity. It defines continuity as a function being continuous at a point if the limit exists and is equal to the function value. Discontinuities can occur if a limit does not exist or is infinite. The chapter applies limits and continuity to solve inequalities involving polynomials and rational functions. Examples show how to use the definition of a limit to evaluate various types of limits and test continuity.
1) This document discusses double-angle and half-angle formulas for trigonometric functions like sine, cosine, and tangent.
2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
1) One-sided limits describe the value a function approaches as the input gets closer to a number from the left or right.
2) The limit of a function exists if and only if the one-sided limits are equal as the input approaches the number.
3) Limits at infinity describe the value a function approaches as the input increases without bound toward positive or negative infinity.
This document discusses properties of logarithms, including:
1) Logarithms with the same base "undo" each other according to the inverse function relationship between logarithms and exponents.
2) Logarithmic expressions can be expanded using properties to write them as sums or differences of individual logarithmic terms, or condensed into a single logarithm.
3) The change of base formula allows converting between logarithms with different bases, with common uses being to change to base 10 or the base of natural logarithms.
Introduction to Sets and Set Operations. The presentation include contents of a KWLH Chart, Essential Questions, and Self-Assessment Questions. With exploration and formative assessments.
An identity is a statement that two trigonometric expressions are equal for every value of the variable. Identities can be verified by manipulating one side of the equation using algebraic substitutions and trigonometric identities until it matches the other side, without moving terms across the equal sign. Practice is important to get better at verifying identities, as each one may require a different approach. Students should keep trying different methods and not get discouraged if it takes time to solve an identity.
1) The document discusses normal subgroups, providing examples and theorems about their properties.
2) A subgroup H of a group G is normal if aH=Ha for all a in G. Some key results are that subgroups of abelian groups and subgroups of index 2 are always normal.
3) The document provides examples of normal subgroups, such as the special linear group SLn,R being normal in the general linear group GLn,R. It also gives a counterexample to show not all groups where every subgroup is normal must be abelian.
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
- Functions can be represented by ordered pairs, graphs, or equations
- The vertical line test identifies functions as those where a vertical line intersects the graph at most once
- Intercepts occur where the graph crosses the x or y-axis
This document summarizes Chapter 10 from a mathematics textbook. The chapter covers limits and continuity. It introduces limits, such as one-sided limits and limits at infinity. It defines continuity as a function being continuous at a point if the limit exists and is equal to the function value. Discontinuities can occur if a limit does not exist or is infinite. The chapter applies limits and continuity to solve inequalities involving polynomials and rational functions. Examples show how to use the definition of a limit to evaluate various types of limits and test continuity.
1) This document discusses double-angle and half-angle formulas for trigonometric functions like sine, cosine, and tangent.
2) It derives formulas that relate trig functions of double and half angles to trig functions of the original angle, such as sin(2x) = 2sin(x)cos(x) and sin(x/2) = ±√(1-cos(x))/2.
3) Examples are provided to demonstrate applying these formulas to simplify trigonometric expressions and derive new identities.
1) One-sided limits describe the value a function approaches as the input gets closer to a number from the left or right.
2) The limit of a function exists if and only if the one-sided limits are equal as the input approaches the number.
3) Limits at infinity describe the value a function approaches as the input increases without bound toward positive or negative infinity.
This document discusses properties of logarithms, including:
1) Logarithms with the same base "undo" each other according to the inverse function relationship between logarithms and exponents.
2) Logarithmic expressions can be expanded using properties to write them as sums or differences of individual logarithmic terms, or condensed into a single logarithm.
3) The change of base formula allows converting between logarithms with different bases, with common uses being to change to base 10 or the base of natural logarithms.
Introduction to Sets and Set Operations. The presentation include contents of a KWLH Chart, Essential Questions, and Self-Assessment Questions. With exploration and formative assessments.
An identity is a statement that two trigonometric expressions are equal for every value of the variable. Identities can be verified by manipulating one side of the equation using algebraic substitutions and trigonometric identities until it matches the other side, without moving terms across the equal sign. Practice is important to get better at verifying identities, as each one may require a different approach. Students should keep trying different methods and not get discouraged if it takes time to solve an identity.
1) The document discusses normal subgroups, providing examples and theorems about their properties.
2) A subgroup H of a group G is normal if aH=Ha for all a in G. Some key results are that subgroups of abelian groups and subgroups of index 2 are always normal.
3) The document provides examples of normal subgroups, such as the special linear group SLn,R being normal in the general linear group GLn,R. It also gives a counterexample to show not all groups where every subgroup is normal must be abelian.
This document provides information about sequences and series in mathematics. It defines sequences, limits of sequences, convergence and divergence of sequences, infinite series, tests to determine convergence of series like the divergence test, limit comparison test, ratio test, root test, and power series. Examples of applying these concepts to specific series are also included.
The order of the given matrix is 2×3. So the maximum no. of elements is 2×3 = 6.
The correct option is B.
The element a32 belongs to 3rd row and 2nd column.
The correct option is B.
3. A matrix whose each diagonal element is unity and all other elements are zero is called
A) Identity matrix B) Unit matrix C) Scalar matrix D) Diagonal matrix
4. A matrix whose each row sums to unity is called
A) Row matrix B) Column matrix C) Unit matrix D) Stochastic matrix
5. The sum of all the elements on the principal diagonal of a square
This document discusses how to solve absolute value inequalities by:
1) Determining whether the absolute value is greater than or less than the variable, which indicates a disjunction or conjunction graph.
2) Solving the inequality for both possibilities of the expression inside the absolute value being positive or negative.
3) Combining the solutions from both possibilities using the appropriate inequality symbol (>, <, etc.) to obtain the final solution set.
1. The document introduces vectors and matrices as ways to collectively represent multiple quantities or relationships between quantities.
2. Vectors are used to represent positions, food orders, prices, and other grouped data. Matrices are used to represent ingredient amounts for different foods and connections between rooms in a floorplan.
3. All of the examples can be expressed using vectors and matrices, with the key information being the numbers in the vectors and matrices.
Roots of real numbers and radical expressionsJessica Garcia
The document defines nth roots and radical expressions. It discusses:
- The definition of an nth root as the value a such that an = b for real numbers a, b and positive integer n.
- Notation for radicals including the index, radical sign, and radicand.
- Simplifying radicals by finding the value such that the expression under the radical equals the radicand.
- Even and odd nth roots depend on the sign of the radicand, with some having one real root and others having two.
The document discusses various methods for solving inequalities, including:
- Properties for adding, subtracting, multiplying, and dividing terms within an inequality
- Using set-builder and interval notation to describe the solution set of an inequality
- Graphical representations using open and closed circles to indicate whether a number is or isn't part of the solution set
The document provides examples of applying these different techniques to solve specific inequalities.
This document provides instructions for solving systems of equations using elimination. It demonstrates eliminating variables by adding or subtracting equations. Sample systems are worked through, showing the steps of identifying which variable to eliminate, combining the equations accordingly, solving for one variable, then substituting back into the original equations to solve for the other. The solutions are checked in both equations to verify they satisfy the system.
This document discusses binomial expansion, which is a method for expanding binomials like (x + a)^n without lengthy multiplication. It introduces key concepts like Pascal's triangle for finding coefficients and the binomial theorem for determining the general pattern of terms in the expansion. Examples are worked through to demonstrate expanding specific binomials like (2x - 3y)^6 according to this method.
The document discusses the rules for matrix multiplication. It states that two matrices can only be multiplied if the number of columns of the first matrix is equal to the number of rows of the second matrix. It provides examples of multiplying different matrices and explains how to calculate each element of the resulting matrix by taking the dot product of the corresponding row and column. It also gives an example of using matrix multiplication to calculate total sales and revenue from sales data organized in matrices.
Identify basic properties of equations
Solve linear equations
Identify identities, conditional equations, and contradictions
Solve for a specific variable (literal equations)
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
Matrices and their operations were discussed. Key points include:
1) A matrix is a rectangular array of numbers. The order of a m x n matrix refers to its m rows and n columns.
2) Common matrix types include row/column matrices (vectors), square matrices, diagonal matrices, scalar matrices, identity matrices, and zero matrices.
3) Basic matrix operations include addition, subtraction, multiplication by a scalar, transpose, and multiplication. Properties like commutativity, associativity, and distributivity apply.
The document discusses function transformations including shifts, reflections, and stretches/compressions. It defines these transformations and provides examples of how they affect the graph of a function. Specifically, it explains that a shift moves a graph up/down or left/right along an axis, a reflection flips the graph across an axis, and a stretch or compression changes the scale of the graph along an axis. Examples are given of reflecting across the x-axis or y-axis and horizontally or vertically stretching/compressing a function. In the end, students are asked to write equations for specific transformations of a quadratic function.
The document discusses several properties of real numbers including:
1) The closure properties of addition and multiplication - the sum and product of any two real numbers is a real number.
2) Commutative, associative, and distributive properties of addition and multiplication.
3) Identity properties of addition and multiplication with 0 and 1 being the identity elements.
4) Inverse properties of addition and multiplication where adding/multiplying a number and its inverse results in the identity element.
The document discusses determinants and their properties. It defines determinants as representing single numbers obtained by multiplying and adding matrix elements in a special way. It then provides formulas for calculating determinants of matrices of order 1, 2 and 3. It also outlines several properties of determinants, such as how interchanging rows/columns, multiplying rows by constants, and adding rows affects the determinant. Finally, it discusses how determinants are used to determine whether systems of linear equations are consistent or inconsistent.
This document defines exponents and radicals. It discusses exponential notation, zero and negative exponents, and the laws of exponents. It also covers scientific notation, nth roots, rational exponents, and rationalizing the denominator. The objectives are to define integer exponents and exponential notation, zero and negative exponents, identify laws of exponents, write numbers using scientific notation, and define nth roots and rational exponents.
The document discusses properties of real numbers. It defines real numbers and distinguishes between rational and irrational numbers. Rational numbers can be expressed as ratios of integers, while irrational numbers have non-terminating, non-repeating decimals. It also outlines relationships between subsets of real numbers and properties that real number operations satisfy, such as commutativity, associativity, identities, inverses, and distribution.
A quadratic inequality is an inequality involving a quadratic expression, such as ax^2 + bx + c < 0. To solve a quadratic inequality, we first find the solutions to the corresponding equation (set the inequality equal to 0) and then test values on either side of those solutions in the original inequality to determine the solutions to the inequality. The solutions to the inequality will be all values of the variable that satisfy the given relationship.
This document contains definitions, examples, and results related to Cauchy sequences, subsequences, and complete metric spaces. It defines a Cauchy sequence as one where the distances between terms gets arbitrarily small as the sequence progresses. It proves that every convergent real sequence is Cauchy. It also defines subsequences and subsequential limits, and proves properties about them. Finally, it defines a complete metric space as one where every Cauchy sequence converges, and provides examples showing the complex numbers form a complete metric space while some subsets of real numbers do not.
This document provides information about determinants of matrices including:
1) It defines the determinant of a 2x2 matrix as a11a22 - a12a21 and provides patterns for calculating the determinant of a 3x3 matrix.
2) It explains that the determinant of an nxn matrix is the sum of all products of n elements with one from each row and column, with a positive or negative sign depending on the number of upward lines.
3) It describes how to calculate the determinant of a 3x3 matrix using cofactors, which are minors multiplied by positive or negative one. The determinant is then expressed as a sum of entries multiplied by their cofactors.
Chapter 3: Linear Systems and Matrices - Part 3/SlidesChaimae Baroudi
The document discusses determinants of matrices. Some key points:
- The determinant (det) of a square matrix is a single number that can be used to determine properties of the matrix, such as invertibility.
- Formulas are given for calculating the determinant of matrices based on their size, such as the cofactor expansion method.
- Certain types of matrices have simple determinant values, such as triangular and diagonal matrices. The determinant of a triangular matrix is the product of its diagonal entries, and the determinant of a diagonal matrix is the product of its diagonal entries.
This document provides information about sequences and series in mathematics. It defines sequences, limits of sequences, convergence and divergence of sequences, infinite series, tests to determine convergence of series like the divergence test, limit comparison test, ratio test, root test, and power series. Examples of applying these concepts to specific series are also included.
The order of the given matrix is 2×3. So the maximum no. of elements is 2×3 = 6.
The correct option is B.
The element a32 belongs to 3rd row and 2nd column.
The correct option is B.
3. A matrix whose each diagonal element is unity and all other elements are zero is called
A) Identity matrix B) Unit matrix C) Scalar matrix D) Diagonal matrix
4. A matrix whose each row sums to unity is called
A) Row matrix B) Column matrix C) Unit matrix D) Stochastic matrix
5. The sum of all the elements on the principal diagonal of a square
This document discusses how to solve absolute value inequalities by:
1) Determining whether the absolute value is greater than or less than the variable, which indicates a disjunction or conjunction graph.
2) Solving the inequality for both possibilities of the expression inside the absolute value being positive or negative.
3) Combining the solutions from both possibilities using the appropriate inequality symbol (>, <, etc.) to obtain the final solution set.
1. The document introduces vectors and matrices as ways to collectively represent multiple quantities or relationships between quantities.
2. Vectors are used to represent positions, food orders, prices, and other grouped data. Matrices are used to represent ingredient amounts for different foods and connections between rooms in a floorplan.
3. All of the examples can be expressed using vectors and matrices, with the key information being the numbers in the vectors and matrices.
Roots of real numbers and radical expressionsJessica Garcia
The document defines nth roots and radical expressions. It discusses:
- The definition of an nth root as the value a such that an = b for real numbers a, b and positive integer n.
- Notation for radicals including the index, radical sign, and radicand.
- Simplifying radicals by finding the value such that the expression under the radical equals the radicand.
- Even and odd nth roots depend on the sign of the radicand, with some having one real root and others having two.
The document discusses various methods for solving inequalities, including:
- Properties for adding, subtracting, multiplying, and dividing terms within an inequality
- Using set-builder and interval notation to describe the solution set of an inequality
- Graphical representations using open and closed circles to indicate whether a number is or isn't part of the solution set
The document provides examples of applying these different techniques to solve specific inequalities.
This document provides instructions for solving systems of equations using elimination. It demonstrates eliminating variables by adding or subtracting equations. Sample systems are worked through, showing the steps of identifying which variable to eliminate, combining the equations accordingly, solving for one variable, then substituting back into the original equations to solve for the other. The solutions are checked in both equations to verify they satisfy the system.
This document discusses binomial expansion, which is a method for expanding binomials like (x + a)^n without lengthy multiplication. It introduces key concepts like Pascal's triangle for finding coefficients and the binomial theorem for determining the general pattern of terms in the expansion. Examples are worked through to demonstrate expanding specific binomials like (2x - 3y)^6 according to this method.
The document discusses the rules for matrix multiplication. It states that two matrices can only be multiplied if the number of columns of the first matrix is equal to the number of rows of the second matrix. It provides examples of multiplying different matrices and explains how to calculate each element of the resulting matrix by taking the dot product of the corresponding row and column. It also gives an example of using matrix multiplication to calculate total sales and revenue from sales data organized in matrices.
Identify basic properties of equations
Solve linear equations
Identify identities, conditional equations, and contradictions
Solve for a specific variable (literal equations)
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
Matrices and their operations were discussed. Key points include:
1) A matrix is a rectangular array of numbers. The order of a m x n matrix refers to its m rows and n columns.
2) Common matrix types include row/column matrices (vectors), square matrices, diagonal matrices, scalar matrices, identity matrices, and zero matrices.
3) Basic matrix operations include addition, subtraction, multiplication by a scalar, transpose, and multiplication. Properties like commutativity, associativity, and distributivity apply.
The document discusses function transformations including shifts, reflections, and stretches/compressions. It defines these transformations and provides examples of how they affect the graph of a function. Specifically, it explains that a shift moves a graph up/down or left/right along an axis, a reflection flips the graph across an axis, and a stretch or compression changes the scale of the graph along an axis. Examples are given of reflecting across the x-axis or y-axis and horizontally or vertically stretching/compressing a function. In the end, students are asked to write equations for specific transformations of a quadratic function.
The document discusses several properties of real numbers including:
1) The closure properties of addition and multiplication - the sum and product of any two real numbers is a real number.
2) Commutative, associative, and distributive properties of addition and multiplication.
3) Identity properties of addition and multiplication with 0 and 1 being the identity elements.
4) Inverse properties of addition and multiplication where adding/multiplying a number and its inverse results in the identity element.
The document discusses determinants and their properties. It defines determinants as representing single numbers obtained by multiplying and adding matrix elements in a special way. It then provides formulas for calculating determinants of matrices of order 1, 2 and 3. It also outlines several properties of determinants, such as how interchanging rows/columns, multiplying rows by constants, and adding rows affects the determinant. Finally, it discusses how determinants are used to determine whether systems of linear equations are consistent or inconsistent.
This document defines exponents and radicals. It discusses exponential notation, zero and negative exponents, and the laws of exponents. It also covers scientific notation, nth roots, rational exponents, and rationalizing the denominator. The objectives are to define integer exponents and exponential notation, zero and negative exponents, identify laws of exponents, write numbers using scientific notation, and define nth roots and rational exponents.
The document discusses properties of real numbers. It defines real numbers and distinguishes between rational and irrational numbers. Rational numbers can be expressed as ratios of integers, while irrational numbers have non-terminating, non-repeating decimals. It also outlines relationships between subsets of real numbers and properties that real number operations satisfy, such as commutativity, associativity, identities, inverses, and distribution.
A quadratic inequality is an inequality involving a quadratic expression, such as ax^2 + bx + c < 0. To solve a quadratic inequality, we first find the solutions to the corresponding equation (set the inequality equal to 0) and then test values on either side of those solutions in the original inequality to determine the solutions to the inequality. The solutions to the inequality will be all values of the variable that satisfy the given relationship.
This document contains definitions, examples, and results related to Cauchy sequences, subsequences, and complete metric spaces. It defines a Cauchy sequence as one where the distances between terms gets arbitrarily small as the sequence progresses. It proves that every convergent real sequence is Cauchy. It also defines subsequences and subsequential limits, and proves properties about them. Finally, it defines a complete metric space as one where every Cauchy sequence converges, and provides examples showing the complex numbers form a complete metric space while some subsets of real numbers do not.
This document provides information about determinants of matrices including:
1) It defines the determinant of a 2x2 matrix as a11a22 - a12a21 and provides patterns for calculating the determinant of a 3x3 matrix.
2) It explains that the determinant of an nxn matrix is the sum of all products of n elements with one from each row and column, with a positive or negative sign depending on the number of upward lines.
3) It describes how to calculate the determinant of a 3x3 matrix using cofactors, which are minors multiplied by positive or negative one. The determinant is then expressed as a sum of entries multiplied by their cofactors.
Chapter 3: Linear Systems and Matrices - Part 3/SlidesChaimae Baroudi
The document discusses determinants of matrices. Some key points:
- The determinant (det) of a square matrix is a single number that can be used to determine properties of the matrix, such as invertibility.
- Formulas are given for calculating the determinant of matrices based on their size, such as the cofactor expansion method.
- Certain types of matrices have simple determinant values, such as triangular and diagonal matrices. The determinant of a triangular matrix is the product of its diagonal entries, and the determinant of a diagonal matrix is the product of its diagonal entries.
The document discusses finding a 3x3 magic square where all entries are distinct perfect squares. It begins by defining a magic square and magic square of squares. It then sets up equations relating the entries and defines how even/odd values will be represented. The majority of the document considers different cases for arranging the first entry in each row and shows through equations that no possible combination can satisfy all constraints, proving a magic square of squares does not exist.
1) A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are specified by the number of rows and columns.
2) The inverse of a square matrix A exists if and only if the determinant of A is not equal to 0. The inverse of A, denoted A^-1, is the matrix that satisfies AA^-1 = A^-1A = I, where I is the identity matrix.
3) For two matrices A and B to be inverses, their product must result in the identity matrix regardless of order, i.e. AB = BA = I. This shows that one matrix undoes the effect of the other.
- There will be no class on Monday for Martin Luther King Day.
- Quiz 1 will be held in class on Wednesday and will cover sections 1.1, 1.2, and 1.3.
- Students should know all definitions clearly for the quiz, which will focus on conceptual understanding rather than lengthy calculations.
This document provides an introduction to basic matrix theory concepts. It defines what a matrix is, explains how to represent vectors as matrices, and covers key matrix concepts like the diagonal matrix, unit matrix, zero matrix, and transpose. It also demonstrates how to add, subtract, and multiply matrices by following specific rules like multiplying rows by columns. Worked examples are provided for adding, subtracting, multiplying, and transposing matrices as well as finding products of matrix operations.
This document contains a 3-page excerpt from the textbook "Elementary Mathematics" by W W L Chen and X T Duong. The excerpt discusses basic algebra concepts including:
- The real number system and subsets like natural numbers, integers, rational numbers, irrational numbers
- Rules of arithmetic operations like addition, subtraction, multiplication, division
- Properties of square roots
- Distributive laws for multiplication
- Arithmetic of fractions including addition and subtraction of fractions.
This document discusses determinants and their properties. It begins by introducing determinants as functions that associate a unique number (real or complex) to square matrices. It then provides examples of calculating determinants of matrices of order 1, 2 and 3 by expanding along rows or columns. The key properties discussed are that the determinant remains unchanged under row/column interchange and that the determinant of a scalar multiple of a matrix is the scalar raised to the power of the matrix order times the determinant of the original matrix. Examples are provided to illustrate these concepts.
Some types of matrices, Eigen value , Eigen vector, Cayley- Hamilton Theorem & applications, Properties of Eigen values, Orthogonal matrix , Pairwise orthogonal, orthogonal transformation of symmetric matrix, denationalization of a matrix by orthogonal transformation (or) orthogonal deduction, Quadratic form and Canonical form , conversion from Quadratic to Canonical form, Order, Index Signature, Nature of canonical form.
This document provides an overview of indices and logarithms in elementary mathematics. It begins by defining integer indices and establishing laws for integer indices. It then extends the definition of indices to rational numbers by defining qth roots. Laws of indices are generalized to apply to rational exponents. Examples are provided to illustrate working with rational exponents. The chapter then introduces exponential functions, defining them as continuous functions that pass through the points (k, ak) for rational k. Laws for exponential functions are stated for real exponents.
A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are written as the number of rows x the number of columns. Each individual entry in the matrix is named by its position, using the matrix name and row and column numbers. Matrices can represent systems of equations or points in a plane. Operations on matrices include addition, multiplication by scalars, and dilation of points represented by matrices.
This document provides an introduction to matrices and their arithmetic operations. It defines what a matrix is, with m rows and n columns. It introduces basic matrix operations like addition, which is done element-wise, and multiplication by scalars. Matrix multiplication is defined as the sum of the products of corresponding entries of the first matrix's rows and second matrix's columns. Several examples are provided to illustrate these matrix operations.
The document discusses various types of matrices:
- Row and column matrices are matrices with only one row or column respectively.
- A square matrix has the same number of rows and columns.
- A diagonal matrix has non-zero elements only along its main diagonal.
- An identity matrix has ones along its main diagonal and zeros elsewhere.
- A scalar matrix has all elements along its main diagonal multiplied by a scalar.
- A null matrix has all elements equal to zero.
The document also discusses properties such as the transpose of a matrix, symmetric matrices, and how to add, subtract and multiply matrices.
The document discusses summation notation and various summation formulas and properties. It defines summation as the sum of all terms in an infinite sequence, represented by ∑. Some key points summarized:
1. The sum of the first n terms of a geometric progression, where the ratio r is between -1 and 1, is Sn = a/(1-r).
2. Common properties of infinite sums include: the sum of two infinite sums is equal to the sum of the individual sums, and the sum of a constant multiplied by terms of an infinite sum is equal to the constant multiplied by the sum.
3. The sum of the first n positive integers can be represented using the formula ∑i=1n
The document discusses the NP-hard Max Cut problem and provides a reduction from the NP-hard NAE-3-SAT problem to Max Cut to prove that Max Cut is also NP-hard. The reduction works by mapping clauses in a NAE-3-SAT instance to a graph instance of Max Cut, such that a solution to one problem can be translated to a solution for the other problem in polynomial time. This shows that any polynomial time algorithm for Max Cut could also be used to solve NAE-3-SAT in polynomial time. The document then provides a simple randomized approximation algorithm for Max Cut that runs in linear time.
A parallelogram is a quadrilateral where opposite sides are equal and opposite angles are equal, and a diagonal of a parallelogram divides it into two congruent triangles.
The document discusses matrices and their properties. It begins by defining a matrix as an ordered rectangular array of numbers or functions. It then discusses the order of a matrix, types of matrices including column, row, square, diagonal and identity matrices. It also defines equality of matrices. The key points are that a matrix represents tabular data, its order is given by the number of rows and columns, and there are different types of matrices based on their properties.
The document provides information about matrix operations and properties. It defines what a matrix is and different types of matrices. It then discusses operations like addition, subtraction, multiplication of matrices. It also covers properties such as transpose, inverse, adjoint and determinant of a matrix. It provides examples to illustrate matrix operations and properties such as finding the inverse and determinant of given matrices.
1. A complex number λ is an eigenvalue of a matrix A if there exists a non-zero vector x such that Ax = λx.
2. If a matrix has complex eigenvalues, it provides important information about the matrix, such as in problems involving vibrations and rotations in space.
3. For a complex eigenvalue λ = a + bi, a is called the real part and b is called the imaginary part. The absolute value |λ| represents the "length" or magnitude of the eigenvalue.
The document contains announcements about an exam, practice exam, review sessions, and exam grading for a class. It states that Exam 2 will be on Thursday, February 25 in class. A practice exam will be uploaded by 2 pm that day. Optional review topics will be covered the next day but will not be on the exam. A review session will be held on Wednesday with office hours from 1-4 pm. It also reminds students that a different class starts on Monday and to collect graded exams on Friday between 7 am and 6 pm.
1. There will be a quiz on Quiz 4 after the next lecture. Exam 2 will be on Feb 25 and cover material from Exam 1 to what is covered on Feb 22.
2. A practice exam will be uploaded on Feb 22 after the remaining material is covered. Optional topics on Feb 23 will not be covered on the exam.
3. Review session on Feb 24 in class. Office hours on Feb 24 from 1-4pm.
- Quiz 4 will be tomorrow covering sections 3.3, 5.1, and 5.2 of the textbook. It will include 3 problems on Cramer's rule, finding eigenvectors given eigenvalues, and finding characteristic polynomials/eigenvalues of 2x2 and 3x3 matrices. Students must show all work.
- Chapter 6 objectives include extending geometric concepts like length, distance, and perpendicularity to Rn. These concepts are useful for least squares fitting of experimental data to a system of equations.
- The inner product of two vectors u and v in Rn is defined as their dot product, which is the sum of the component-wise products of corresponding elements in u and v.
1. Quiz 4 will cover sections 3.3, 5.1, and 5.2 and will be on Thursday, February 18.
2. To find the nth power of a matrix A that has been diagonalized as A = PDP-1, one raises the diagonal elements of D to the nth power to obtain Dn, leaving P and P-1 unchanged.
3. A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, allowing it to be written as A = PDP-1, where the columns of P are the eigenvectors and the diagonal elements of D are the corresponding eigenvalues.
1. The document announces that students should bring any exam 1 grade questions without delay, and that the homework for exam 2 has been uploaded and may be updated. It also notes that the last day to drop the class is February 4th and there is no class on that date.
2. The document covers topics from the last class including computing 3x3 determinants, determinants of triangular matrices, and techniques for larger matrices.
3. The document then provides examples of computing determinants and discusses important properties including that row operations do not change the determinant value while row interchanges flip the sign, and multiplying a row scales the determinant.
The document discusses the process for finding the eigenvalues of a square matrix. It begins by defining the characteristic equation as det(A - λI) = 0, where A is the matrix and λI subtracts λ from the diagonal. The characteristic polynomial is obtained by computing this determinant. For a 2x2 matrix, it is a quadratic equation that can be factored to find the two eigenvalues. Larger matrices may require numerical methods. The sum of eigenvalues equals the trace, and their product equals the determinant. A matrix will always have n eigenvalues for its size n. An example problem is presented to demonstrate the full process.
1. The matrix is not invertible as it has repeated rows.
2. The eigenvalue is 0 since a matrix is not invertible if it has 0 as an eigenvalue.
3. The eigenvectors corresponding to 0 can be found by reducing the matrix A - 0I to row echelon form. This gives the equation x1 + x2 + x3 = 0 with x2 and x3 as free variables, so two linearly independent eigenvectors are (1, -1, 0) and (1, 0, -1).
Eigenvalues and Eigenvectors (Tacoma Narrows Bridge video included)Prasanth George
- There is a quiz tomorrow on sections 3.1 and 3.2 of the course material. Calculators will not be allowed and determinants must be calculated using the methods learned.
- Eigenvalues and eigenvectors are related to the linear transformation of a matrix A acting on a vector x. They give a better understanding of the transformation.
- The 1940 collapse of the Tacoma Narrows Bridge is explained by oscillations caused by the wind frequency matching the bridge's natural frequency, which is the eigenvalue of smallest magnitude based on a mathematical model of the bridge. Eigenvalues are important for engineering structure design.
1. Quiz 3 will cover sections 3.1 and 3.2 on February 11th. No calculators will be allowed and determinants must be found using the methods taught.
2. The homework problems have been updated, so students should check for the latest list.
3. To find the inverse of a 3x3 matrix A, first find the adjugate of A (denoted adjA) by writing the cofactors with alternating signs, then divide adjA by the determinant of A.
The document contains announcements for an upcoming exam:
1. Students should bring any grade related questions about quiz 2 without delay. Test 1 will be on February 1st covering sections 1.1-1.5, 1.7-1.8, 2.1-2.3 and 2.8-2.9.
2. A sample exam 1 will be posted by that evening. Students should review for the exam after the lecture.
3. The instructor will be available in their office all day the following day to answer any questions.
It also provides tips for preparing for the exam, including doing homework problems and sample exams within the time limit to practice time management.
The document contains notes from a previous linear algebra class covering the following topics:
1. There will be a quiz tomorrow on sections 1.1-1.3 focusing on concepts rather than lengthy calculations.
2. Previous topics included systems of linear equations, row reduction, pivot positions, basic and free variables, and the span of vectors.
3. Determining if a vector is in the span of other vectors is equivalent to checking if the corresponding linear system is consistent.
4. Examples are provided of determining if homogeneous systems have non-trivial solutions based on the presence of free variables. The general solution of a homogeneous system is expressed in parametric vector form.
The document contains announcements and information about an upcoming exam:
- A quiz and test are scheduled. Sample exams and review sessions will be provided.
- Exam 1 will cover several sections of the textbook and the professor will be available for questions.
- Tips are provided for studying including doing homework, examples, and practicing sample exams.
- Sections about subspaces and column/null spaces of matrices are summarized, including properties and examples.
Quiz 2 will be held on January 27 covering sections 1.4, 1.5, 1.7, and 1.8. Test 1 is scheduled for February 1. The document then provides steps to find the inverse of a 2x2 matrix, discusses invertibility if the determinant is 0, and gives an example of finding the inverse of a 3x3 matrix using row reduction of the augmented matrix.
The document discusses the following:
1. There will be a quiz on Jan 27 covering sections 1.4, 1.5, 1.7, and 1.8 and any issues with quiz 1 should be discussed asap.
2. Test 1 will be on Feb 1 in class with more details to come.
3. Matrix multiplication is defined only when the number of columns of the first matrix equals the number of rows of the second matrix.
Quiz 2 will cover sections 1.4, 1.5, 1.7, and 1.8 on Wednesday January 27. Students with issues on quiz 1 should discuss with the instructor as soon as possible. The solution to quiz 1 will be posted on the website by Monday.
The document discusses linear transformations and provides examples of applying linear transformations to vectors. It defines key concepts such as the domain, co-domain, and range of a transformation. Examples are provided of interesting linear transformations including rotation and reflection transformations. Solutions to examples involving finding the image of vectors under given linear transformations are shown.
The document discusses linear transformations and linear independence. It contains examples and explanations of:
1) How a matrix A can transform a vector x from R4 to a new vector b in R2, representing the linear transformation.
2) How finding vectors x such that Ax=b is equivalent to finding pre-images of b under the transformation A.
3) Key concepts related to linear transformations like domain and range.
The document contains announcements and information about a class. It announces corrections to lecture slides, the last day to drop the class with a refund, and provides definitions and examples related to echelon form, reduced row echelon form, pivot positions, and solving systems of linear equations.
The document contains announcements from a class instructor. It notifies students that if they have not been able to access the class website or did not receive an email, to contact the instructor. It also reminds students that homeworks are posted on the class website and to check for any updates.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
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How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
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How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
1. Announcements
Please bring any grade related questions regarding exam 1
without delay.
Homework set for exam 2 has been uploaded. Please check it
often, I may make small inclusions/exclusions.
Planning to do parts of chapters 3, 5 and 6 for exam 2.
Last day to drop this class with grade "W" is Feb 4.
2. Section 3.1 Introduction to Determinants
1. A 2 × 2 matrix A is invertible if and only if det A= 0.
3. Section 3.1 Introduction to Determinants
1. A 2 × 2 matrix A is invertible if and only if det A= 0.
2. We can now extend this idea to a 3 × 3 or larger matrices.
4. Section 3.1 Introduction to Determinants
1. A 2 × 2 matrix A is invertible if and only if det A= 0.
2. We can now extend this idea to a 3 × 3 or larger matrices.
3. Determinants exist only for square matrices.
Notation The notation aij means the element in the i -th row and
j -th column of a matrix.
5. Section 3.1 Introduction to Determinants
1. A 2 × 2 matrix A is invertible if and only if det A= 0.
2. We can now extend this idea to a 3 × 3 or larger matrices.
3. Determinants exist only for square matrices.
Notation The notation aij means the element in the i -th row and
j -th column of a matrix.
So a23 means the element in the second row, third column of a
given matrix.
7. Determinant of a 2 × 2 matrix
You know this one!!
If
a11 a12
A =
a21 a22
Here, det A =a11 a22 − a21 a12 . It is a number.
8. What about a 3 × 3 matrix?
If
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
9. What about a 3 × 3 matrix?
If
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
We have to break this down to multiple 2 × 2 determinants.
10. What about a 3 × 3 matrix?
You can start the computation using any row or column as an
anchor.
11. What about a 3 × 3 matrix?
You can start the computation using any row or column as an
anchor.
Suppose you choose the rst row.
Each entry of the rst row will give one term each as follows.
12. What about a 3 × 3 matrix?
You can start the computation using any row or column as an
anchor.
Suppose you choose the rst row.
Each entry of the rst row will give one term each as follows.
Add the terms at the end to get det A.
13. What about a 3 × 3 matrix?
You can start the computation using any row or column as an
anchor.
Suppose you choose the rst row.
Each entry of the rst row will give one term each as follows.
Add the terms at the end to get det A.
To get the rst term of det A, cover the row and column
corresponding to a11 .
14. What about a 3 × 3 matrix?
You can start the computation using any row or column as an
anchor.
Suppose you choose the rst row.
Each entry of the rst row will give one term each as follows.
Add the terms at the end to get det A.
To get the rst term of det A, cover the row and column
corresponding to a11 .
a11 a12 a13
a21 a22 a23
a31 a32 a33
15. Multiply a11 with the determinant of the remaining matrix
a22 a23
a32 a33
16. Multiply a11 with the determinant of the remaining matrix
a22 a23
a32 a33
Thus the rst term is (
a11 a22 a33 − a32 a23 ).
17. Multiply a11 with the determinant of the remaining matrix
a22 a23
a32 a33
Thus the rst term is a11 (a22 a33 − a32 a23 ).
To get the second term of det A, cover the row and column
corresponding to a12 .
a11 a12 a13
a21 a22 a23
a31 a32 a33
18. Multiply a11 with the determinant of the remaining matrix
a22 a23
a32 a33
Thus the rst term is a11 (a22 a33 − a32 a23 ).
To get the second term of det A, cover the row and column
corresponding to a12 .
a11 a12 a13
a21 a22 a23
a31 a32 a33
Multiply the negative of a12 with the determinant of the
remaining matrix
a21 a23
a31 a33
20. Thus the second term is −a12 (a21 a33 − a31 a23 ).
To get the third term of det A, cover the row and column
corresponding to a13 .
a11 a12 a13
a21 a22 a23
a31 a32 a33
21. Thus the second term is −a12 (a21 a33 − a31 a23 ).
To get the third term of det A, cover the row and column
corresponding to a13 .
a11 a12 a13
a21 a22 a23
a31 a32 a33
Multiply a13 with the determinant of the remaining matrix
a21 a22
a31 a32
22. Thus the second term is −a12 (a21 a33 − a31 a23 ).
To get the third term of det A, cover the row and column
corresponding to a13 .
a11 a12 a13
a21 a22 a23
a31 a32 a33
Multiply a13 with the determinant of the remaining matrix
a21 a22
a31 a32
Thus the second term is (
a13 a21 a32 − a31 a22 ).
23. Add the 3 terms you obtained above
(
a11 a22 a33 − a32 a23 ) − a12 (a21 a33 − a31 a23 ) + a13 (a21 a32 − a31 a22 )
This is det A for a 3 × 3 matrix A.
24. Add the 3 terms you obtained above
(
a11 a22 a33 − a32 a23 ) − a12 (a21 a33 − a31 a23 ) + a13 (a21 a32 − a31 a22 )
This is det A for a 3 × 3 matrix A.
DO NOT try to memorize this as a formula
25. Add the 3 terms you obtained above
(
a11 a22 a33 − a32 a23 ) − a12 (a21 a33 − a31 a23 ) + a13 (a21 a32 − a31 a22 )
This is det A for a 3 × 3 matrix A.
DO NOT try to memorize this as a formula
Remember the steps (all the covering and multiplying games)!!
26. Add the 3 terms you obtained above
(
a11 a22 a33 − a32 a23 ) − a12 (a21 a33 − a31 a23 ) + a13 (a21 a32 − a31 a22 )
This is det A for a 3 × 3 matrix A.
DO NOT try to memorize this as a formula
Remember the steps (all the covering and multiplying games)!!
To nd determinant of a 4 × 4 matrix A, break it down into
four 3 × 3 determinants using the same idea. (more work).
This method works for a square matrix of any size.
28. FAQs
Which row to choose for anchor? Any row (or column)!!
29. FAQs
Which row to choose for anchor? Any row (or column)!!
Any caveats?? Yes!! Need to make sure that you do proper
sign alternating depending on which row or column you
choose. Keep the following in mind.
+ − +
A= − + −
+ − +
30. FAQs
Which row to choose for anchor? Any row (or column)!!
Any caveats?? Yes!! Need to make sure that you do proper
sign alternating depending on which row or column you
choose. Keep the following in mind.
+ − +
A= − + −
+ − +
So, if you decide to use second column, the rst term will be
negative, the second positive and the third negative. (with
proper covering and multiplying)
31. FAQs
Which row to choose for anchor? Any row (or column)!!
Any caveats?? Yes!! Need to make sure that you do proper
sign alternating depending on which row or column you
choose. Keep the following in mind.
+ − +
A= − + −
+ − +
So, if you decide to use second column, the rst term will be
negative, the second positive and the third negative. (with
proper covering and multiplying)
Choose a row or column with as many zeros as possible.
32. Before we go further..
Notation: Use a pair of vertical lines for determinants.
33. Before we go further..
Notation: Use a pair of vertical lines for determinants.
Example
If
1 2 3
A= 4 5 6
7 8 9
then
1 2 3
det A = 4 5 6
7 8 9
34. Going back to our 3 × 3 matrix
a11 a12 a13
A = a21 a22 a23 ,
a31 a32 a33
35. Going back to our 3 × 3 matrix
a11 a12 a13
A = a21 a22 a23 ,
a31 a32 a33
we can write
a22 a23 a21 a23 a21 a22
det A = a11 −a12 +a13
a32 a33 a31 a33 a31 a32
C11 C12 C13
36. Going back to our 3 × 3 matrix
a11 a12 a13
A = a21 a22 a23 ,
a31 a32 a33
we can write
a22 a23 a21 a23 a21 a22
det A = a11 −a12 +a13
a32 a33 a31 a33 a31 a32
C11 C12 C13
Here C11 , C12 and C13 are called the cofactors of A.
37. Going back to our 3 × 3 matrix
a11 a12 a13
A = a21 a22 a23 ,
a31 a32 a33
we can write
a22 a23 a21 a23 a21 a22
det A = a11 −a12 +a13
a32 a33 a31 a33 a31 a32
C11 C12 C13
Here C11 , C12 and C13 are called the cofactors of A.
This method of computing determinants is called cofactor
expansion across rst row.
39. In General..
Theorem
1. The determinant of an n ×n matrix A can be computed by
cofactor expansion along any row or column.
40. In General..
Theorem
1. The determinant of an n ×n matrix A can be computed by
cofactor expansion along any row or column.
2. Expansion across the i th row will be
det A = ai 1 Ci 1 + ai 2 Ci 2 + . . . + ain Cin .
Don't forget to take care of proper sign alternations depending
on the row.
41. In General..
Theorem
1. The determinant of an n ×n matrix A can be computed by
cofactor expansion along any row or column.
2. Expansion across the i th row will be
det A = ai 1 Ci 1 + ai 2 Ci 2 + . . . + ain Cin .
Don't forget to take care of proper sign alternations depending
on the row.
3. Expansion across the j th column will be
det A = a1j C1j + a2j C2j + . . . + anj Cnj .
Don't forget to take care of proper sign alternations depending
on the column.
42. Example 2, section 3.1
Compute using cofactor expansion along rst row.
0 5 1
4 −3 0
2 4 1
43. Example 2, section 3.1
Compute using cofactor expansion along rst row.
0 5 1
4 −3 0
2 4 1
Solution:
−3 0
det A = 0
4 1
−3
44. Example 2, section 3.1
Compute using cofactor expansion along rst row.
0 5 1
4 −3 0
2 4 1
Solution:
−3 0 4 0
det A = 0 −5
4 1 2 1
−3 4
45. Example 2, section 3.1
Compute using cofactor expansion along rst row.
0 5 1
4 −3 0
2 4 1
Solution:
−3 0 4 0 4 −3
det A = 0 −5 +1
4 1 2 1 2 4
−3 4 22
56. Comments
1. Again, be careful with the alternating signs.
57. Comments
1. Again, be careful with the alternating signs.
2. If you are expanding down the second column, the rst term
will be negative, second positive (but already we have a -3)
and the third negative.
58. Example 8, section 3.1
Compute using cofactor expansion along rst row.
8 1 6
4 0 3
3 −2 5
59. Example 8, section 3.1
Compute using cofactor expansion along rst row.
8 1 6
4 0 3
3 −2 5
8 1 6
4 0 3
3 −2 5
det A =
60. Example 8, section 3.1
Compute using cofactor expansion along rst row.
8 1 6
4 0 3
3 −2 5
8 1 6
4 0 3
3 −2 5
0 3
det A = 8
−2 5
6
63. Denition
A square matrix A is a Triangular matrix if the entries above OR
below the main diagonal are ALL zeros
64. Denition
A square matrix A is a Triangular matrix if the entries above OR
below the main diagonal are ALL zeros
Theorem
If A is a triangular matrix, then det A is the product of entries on
the main diagonal of A.
67. Larger Convenient Matrices
1. If you have a 4 × 4 or larger matrix with a row or column
mostly zeros, use that row(column) as the anchor.
68. Larger Convenient Matrices
1. If you have a 4 × 4 or larger matrix with a row or column
mostly zeros, use that row(column) as the anchor.
2. Be careful with the sign alterations.
69. Larger Convenient Matrices
1. If you have a 4 × 4 or larger matrix with a row or column
mostly zeros, use that row(column) as the anchor.
2. Be careful with the sign alterations.
3. Have a sign template of proper size handy.
70. Example 10, section 3.1
Compute the following determinant using least amount of
computation.
1 −2 5 2
0 0 3 0
2 −6 −7 5
5 0 4 4
71. Example 10, section 3.1
Compute the following determinant using least amount of
computation.
1 −2 5 2
0 0 3 0
2 −6 −7 5
5 0 4 4
Use row 2 as the anchor. To be sure about the signs use the
following
+ − + −
− + − +
+ − + −
− + − +
72. Example 10, section 3.1
Compute the following determinant using least amount of
computation.
1 −2 5 2
0 0 3 0
2 −6 −7 5
5 0 4 4
Use row 2 as the anchor. To be sure about the signs use the
following
+ − + −
− + − +
+ − + −
− + − +
Only the cofactor of 3 matters here. It will be negative. Others are
all zero.
80. Sarrus' Mnemonic Rule
1. An easy to remember method for 3 × 3 matrices
2. DO NOT apply this method for larger matrices.
3. Make sure all rows and columns are properly aligned, otherwise
it becomes very confusing.
81. Sarrus' Mnemonic Rule
1. An easy to remember method for 3 × 3 matrices
2. DO NOT apply this method for larger matrices.
3. Make sure all rows and columns are properly aligned, otherwise
it becomes very confusing.
4. Start by repeating the rst 2 rows immediately beneath the
determinant.