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.
This document provides an introduction to basic matrix operations including addition, subtraction, scalar multiplication, and matrix multiplication. It defines matrices and how to perform basic arithmetic on matrices. It also introduces the concept of matrix equations and using matrix multiplication to solve systems of linear equations. Key points covered include:
- How to add and subtract matrices by adding or subtracting the corresponding entries
- Scalar multiplication involves multiplying each entry of a matrix by a scalar number
- Matrix multiplication involves multiplying rows of one matrix with columns of another, with the constraint that the number of columns of the first matrix must equal the number of rows of the second matrix.
- Matrix multiplication is not commutative in general.
This document provides definitions and explanations of various mathematical concepts. It defines linear and simultaneous equations, quadratic equations, matrices including types of matrices. It also discusses sequences and series, percentages, discounts, commission, and interest. Key terms are defined such as determinant, properties of addition and multiplication for matrices, and formulas for arithmetic and geometric progressions.
The document discusses matrix algebra and operations on matrices. It defines a matrix as a rectangular table of numbers with rows and columns. A matrix with R rows and C columns is denoted as an R x C matrix. Individual entries in a matrix are denoted by their row and column position, such as a32 for the entry in the 3rd row and 2nd column. There are two main types of operations on matrices - adding/subtracting same-sized matrices entry by entry, and multiplying matrices. Matrix multiplication involves multiplying corresponding entries of a row and column and summing the products.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
Please go through the slides. It is very interesting way to learn this chapter for 2020-21.If you like this PPT please put a thanks message in my number 9826371828.
The document provides an overview of a lecture covering matrices, matrix algebra, vectors, homogeneous coordinates, and transformations in homogeneous coordinates. Key points include: matrices are arrays of numbers; operations on matrices include addition, multiplication by a scalar, and multiplication; vectors can represent points in space as column or row matrices; homogeneous coordinates allow points to be represented by 4D vectors, enabling translations and other transformations to be described by 4x4 matrices. This provides a unified approach for combining multiple transformations.
The document discusses matrices and their operations. It defines what a matrix is, provides examples of different types of matrices, and covers key matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. It also defines important matrix concepts such as the transpose of a matrix, inverse of a matrix, and properties related to these operations and concepts.
This document provides an introduction to basic matrix operations including addition, subtraction, scalar multiplication, and matrix multiplication. It defines matrices and how to perform basic arithmetic on matrices. It also introduces the concept of matrix equations and using matrix multiplication to solve systems of linear equations. Key points covered include:
- How to add and subtract matrices by adding or subtracting the corresponding entries
- Scalar multiplication involves multiplying each entry of a matrix by a scalar number
- Matrix multiplication involves multiplying rows of one matrix with columns of another, with the constraint that the number of columns of the first matrix must equal the number of rows of the second matrix.
- Matrix multiplication is not commutative in general.
This document provides definitions and explanations of various mathematical concepts. It defines linear and simultaneous equations, quadratic equations, matrices including types of matrices. It also discusses sequences and series, percentages, discounts, commission, and interest. Key terms are defined such as determinant, properties of addition and multiplication for matrices, and formulas for arithmetic and geometric progressions.
The document discusses matrix algebra and operations on matrices. It defines a matrix as a rectangular table of numbers with rows and columns. A matrix with R rows and C columns is denoted as an R x C matrix. Individual entries in a matrix are denoted by their row and column position, such as a32 for the entry in the 3rd row and 2nd column. There are two main types of operations on matrices - adding/subtracting same-sized matrices entry by entry, and multiplying matrices. Matrix multiplication involves multiplying corresponding entries of a row and column and summing the products.
This presentation describes Matrices and Determinants in detail including all the relevant definitions with examples, various concepts and the practice problems.
Please go through the slides. It is very interesting way to learn this chapter for 2020-21.If you like this PPT please put a thanks message in my number 9826371828.
The document provides an overview of a lecture covering matrices, matrix algebra, vectors, homogeneous coordinates, and transformations in homogeneous coordinates. Key points include: matrices are arrays of numbers; operations on matrices include addition, multiplication by a scalar, and multiplication; vectors can represent points in space as column or row matrices; homogeneous coordinates allow points to be represented by 4D vectors, enabling translations and other transformations to be described by 4x4 matrices. This provides a unified approach for combining multiple transformations.
The document discusses matrices and their operations. It defines what a matrix is, provides examples of different types of matrices, and covers key matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. It also defines important matrix concepts such as the transpose of a matrix, inverse of a matrix, and properties related to these operations and concepts.
class-10 Maths Ncert Book | Chapter-3 | Slides By MANAV |Slides With MANAV
This document discusses representing situations involving two related variables with pairs of linear equations and solving them graphically. It provides three examples:
1) A girl spends money at a fair based on rides and a game. This is represented as two intersecting lines, with a unique solution of 4 rides and playing the game 2 times.
2) Two friends buy pencils and erasers with different amounts. This is represented by coinciding lines, meaning the equations are equivalent and there are infinitely many solutions.
3) Two rail lines are represented by parallel lines, meaning there is no intersection and no solution to the pair of equations.
The document explains that pairs of linear equations can have a unique solution
The document defines matrices and their properties. It discusses the key concepts of matrices including:
- Types of matrices such as row, column, rectangular, square, null, diagonal, and identity matrices.
- Properties of matrices including order, rows, columns, equality, transpose, addition, multiplication, and inverse.
- Special matrices like symmetric, skew-symmetric, and scalar matrices.
- How to evaluate determinants of square matrices and solve systems of linear equations using matrices.
Matrices
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 30, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
This document provides an introduction to calculus by discussing pure versus applied mathematics. It then reviews basic mathematical concepts such as exponents, algebraic expressions, solving equations, inequalities, and sets that are used in numerical analysis. Finally, it discusses graphical representations of rectangular and polar coordinate systems and includes examples of converting between the two systems.
The document defines and provides examples of different types of matrices, including:
- Square matrices, where the number of rows equals the number of columns.
- Rectangular matrices, where the number of rows does not equal the number of columns.
- Row matrices, with only one row.
- Column matrices, with only one column.
- Null or zero matrices, with all elements equal to zero.
- Diagonal matrices, with all elements equal to zero except those on the main diagonal.
The document also discusses transpose, adjoint, and addition of matrices.
This document provides an overview of matrices including:
- How to describe matrices using m rows and n columns
- Common types of matrices such as row, column, zero, square, diagonal, and unit matrices
- Basic matrix operations including addition, subtraction, scalar multiplication
- Rules for matrix multiplication including that matrices must be conformable
- The transpose of a matrix which is obtained by interchanging rows and columns
- Properties of transposed matrices including (A+B)T = AT + BT and (AB)T = BTAT
This document provides important points and formulas for mathematics. It includes:
1) Definitions and properties of numbers like natural numbers, integers, rational numbers, and irrational numbers.
2) Formulas for topics like percentages, simple interest, compound interest, speed, distance, time, and areas of shapes.
3) Explanations of concepts in algebra like quadratic equations, expansion and factorization of expressions, and ordering.
4) Formulas for area and perimeter of shapes, Pythagoras' theorem, surface area and volume of solids like cylinders and cones.
The document serves as a quick reference guide for key mathematical definitions, properties, formulas, and concepts across various topics
This document provides definitions and properties of matrices and determinants. Some key points include:
- A matrix is a rectangular array of numbers or expressions. The order of a matrix refers to its number of rows and columns.
- Special types of matrices include square, diagonal, identity, and zero matrices. Transpose of a matrix is obtained by interchanging rows and columns.
- Determinants can be calculated for square matrices and have various properties. The adjoint and inverse of a matrix are also defined.
- Systems of linear equations can be solved using matrices by computing the inverse or using properties of determinants.
The document defines key concepts in set theory including:
1. A set is a well-defined collection of distinct objects called elements. Georg Cantor is credited with creating set theory.
2. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties the elements share.
3. Operations on sets include intersection, union, difference, and complement. Intersection is the set of common elements, union is all elements in either set, difference is elements only in the first set, and complement is elements not in the set.
The document discusses differentiation and tangents. It explains that differentiation finds the gradient of a curve at a point and is needed because curves have changing gradients. It provides examples of differentiating simple functions like y=3x^5. Tangents are lines that touch a curve at a single point, with the same gradient as the curve at that point. To find the equation of a tangent, take the derivative and plug in the x-value to get the slope, then use the point to find the y-intercept. Normals are perpendicular to tangents, with a gradient equal to the negative reciprocal of the tangent's gradient.
CLASS 9 LINEAR EQUATIONS IN TWO VARIABLES PPT05092000
This document provides information about linear equations in two variables. It defines linear equations and explains that a linear equation in two variables can be written in the form ax + by = c. The document also discusses finding the solutions of linear equations, graphing linear equations, and equations of lines parallel to the x-axis and y-axis. Examples are provided to illustrate key concepts. In the summary, key points are restated such as linear equations having infinitely many solutions and the graph of a linear equation being a straight line.
This document provides an overview of matrices, including how they are named, the elements within them, and how their dimensions are written. It explains that matrices are arrays of numbers, with elements designated by their row and column position. Square matrices have the same number of rows and columns, while non-square matrices have different numbers of rows and columns. Examples are provided to demonstrate finding specific elements and their locations within matrices of various sizes.
This document discusses linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides examples of linear equations and discusses how to graph them by plotting the x and y intercepts. It also explains how to determine if a given ordered pair is a solution to a linear equation by substituting the x and y values into the equation. Finally, it discusses different methods for solving systems of linear equations, including substitution and elimination.
- The document discusses determinants of square matrices, including how to calculate the determinant of matrices of various orders, properties of determinants, and some applications of determinants.
- Key concepts covered include minors, cofactors, expanding determinants in terms of minors and cofactors, properties such as how determinants change with row/column operations, and using determinants to solve systems of linear equations.
- Examples are provided to demonstrate calculating determinants and using properties to simplify or prove identities about determinants.
BSC_COMPUTER _SCIENCE_UNIT-3_DISCRETE MATHEMATICSRai University
The document provides information about matrices including:
- Definitions of matrix, order of a matrix, types of matrices like row, column, square, null, identity, diagonal, scalar, and transpose matrices.
- Operations on matrices like addition, subtraction, scalar multiplication and matrix multiplication along with their properties.
- Examples of finding the sum of matrices.
- The document discusses various types of matrices like symmetric, skew-symmetric, singular, equal, negative, orthogonal, upper/lower triangular, trace, idempotent, involuntary, conjugate, unitary, Hermitian, skew-Hermitian matrices and minors of a matrix.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
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.
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 discusses matrices and their applications in solving systems of linear equations. It begins with defining what a matrix is and some special types of matrices including square, identity, zero, diagonal, commutative, triangular, and symmetric matrices. It also covers matrix operations like transpose and properties of determinants. The document then discusses using matrices to solve systems of linear equations through homogeneous systems, Gaussian elimination, Gaussian-Jordan elimination, and Cramer's rule. It includes examples and equations to illustrate the concepts and applications of matrices.
The rank of a matrix is the maximum number of linearly independent rows. A matrix has rank 0 only if it is the zero matrix. The inverse of a matrix A, denoted A^-1, is defined such that A*A^-1 = I, the identity matrix. To calculate the inverse, one takes the transpose of the cofactor matrix and divides each element by the determinant of the original matrix.
Matrices and its Applications to Solve Some Methods of Systems of Linear Equa...Rwan Kamal
This document is a research project submitted to the University of Duhok examining matrices and their applications in solving systems of linear equations. The project begins with an introduction to basic matrix concepts such as types of matrices, operations on matrices, and finding the inverse and determinant of matrices. It then discusses how matrices can be used to solve different types of systems of linear equations, including homogeneous systems, Gaussian elimination, Gaussian-Jordan elimination, and Cramer's rule. The document contains acknowledgments, contents, abstract, and introduction sections before presenting the technical matrix concepts and methods for solving systems of linear equations.
class-10 Maths Ncert Book | Chapter-3 | Slides By MANAV |Slides With MANAV
This document discusses representing situations involving two related variables with pairs of linear equations and solving them graphically. It provides three examples:
1) A girl spends money at a fair based on rides and a game. This is represented as two intersecting lines, with a unique solution of 4 rides and playing the game 2 times.
2) Two friends buy pencils and erasers with different amounts. This is represented by coinciding lines, meaning the equations are equivalent and there are infinitely many solutions.
3) Two rail lines are represented by parallel lines, meaning there is no intersection and no solution to the pair of equations.
The document explains that pairs of linear equations can have a unique solution
The document defines matrices and their properties. It discusses the key concepts of matrices including:
- Types of matrices such as row, column, rectangular, square, null, diagonal, and identity matrices.
- Properties of matrices including order, rows, columns, equality, transpose, addition, multiplication, and inverse.
- Special matrices like symmetric, skew-symmetric, and scalar matrices.
- How to evaluate determinants of square matrices and solve systems of linear equations using matrices.
Matrices
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 30, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
This document provides an introduction to calculus by discussing pure versus applied mathematics. It then reviews basic mathematical concepts such as exponents, algebraic expressions, solving equations, inequalities, and sets that are used in numerical analysis. Finally, it discusses graphical representations of rectangular and polar coordinate systems and includes examples of converting between the two systems.
The document defines and provides examples of different types of matrices, including:
- Square matrices, where the number of rows equals the number of columns.
- Rectangular matrices, where the number of rows does not equal the number of columns.
- Row matrices, with only one row.
- Column matrices, with only one column.
- Null or zero matrices, with all elements equal to zero.
- Diagonal matrices, with all elements equal to zero except those on the main diagonal.
The document also discusses transpose, adjoint, and addition of matrices.
This document provides an overview of matrices including:
- How to describe matrices using m rows and n columns
- Common types of matrices such as row, column, zero, square, diagonal, and unit matrices
- Basic matrix operations including addition, subtraction, scalar multiplication
- Rules for matrix multiplication including that matrices must be conformable
- The transpose of a matrix which is obtained by interchanging rows and columns
- Properties of transposed matrices including (A+B)T = AT + BT and (AB)T = BTAT
This document provides important points and formulas for mathematics. It includes:
1) Definitions and properties of numbers like natural numbers, integers, rational numbers, and irrational numbers.
2) Formulas for topics like percentages, simple interest, compound interest, speed, distance, time, and areas of shapes.
3) Explanations of concepts in algebra like quadratic equations, expansion and factorization of expressions, and ordering.
4) Formulas for area and perimeter of shapes, Pythagoras' theorem, surface area and volume of solids like cylinders and cones.
The document serves as a quick reference guide for key mathematical definitions, properties, formulas, and concepts across various topics
This document provides definitions and properties of matrices and determinants. Some key points include:
- A matrix is a rectangular array of numbers or expressions. The order of a matrix refers to its number of rows and columns.
- Special types of matrices include square, diagonal, identity, and zero matrices. Transpose of a matrix is obtained by interchanging rows and columns.
- Determinants can be calculated for square matrices and have various properties. The adjoint and inverse of a matrix are also defined.
- Systems of linear equations can be solved using matrices by computing the inverse or using properties of determinants.
The document defines key concepts in set theory including:
1. A set is a well-defined collection of distinct objects called elements. Georg Cantor is credited with creating set theory.
2. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties the elements share.
3. Operations on sets include intersection, union, difference, and complement. Intersection is the set of common elements, union is all elements in either set, difference is elements only in the first set, and complement is elements not in the set.
The document discusses differentiation and tangents. It explains that differentiation finds the gradient of a curve at a point and is needed because curves have changing gradients. It provides examples of differentiating simple functions like y=3x^5. Tangents are lines that touch a curve at a single point, with the same gradient as the curve at that point. To find the equation of a tangent, take the derivative and plug in the x-value to get the slope, then use the point to find the y-intercept. Normals are perpendicular to tangents, with a gradient equal to the negative reciprocal of the tangent's gradient.
CLASS 9 LINEAR EQUATIONS IN TWO VARIABLES PPT05092000
This document provides information about linear equations in two variables. It defines linear equations and explains that a linear equation in two variables can be written in the form ax + by = c. The document also discusses finding the solutions of linear equations, graphing linear equations, and equations of lines parallel to the x-axis and y-axis. Examples are provided to illustrate key concepts. In the summary, key points are restated such as linear equations having infinitely many solutions and the graph of a linear equation being a straight line.
This document provides an overview of matrices, including how they are named, the elements within them, and how their dimensions are written. It explains that matrices are arrays of numbers, with elements designated by their row and column position. Square matrices have the same number of rows and columns, while non-square matrices have different numbers of rows and columns. Examples are provided to demonstrate finding specific elements and their locations within matrices of various sizes.
This document discusses linear equations in two variables. It defines a linear equation as one that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides examples of linear equations and discusses how to graph them by plotting the x and y intercepts. It also explains how to determine if a given ordered pair is a solution to a linear equation by substituting the x and y values into the equation. Finally, it discusses different methods for solving systems of linear equations, including substitution and elimination.
- The document discusses determinants of square matrices, including how to calculate the determinant of matrices of various orders, properties of determinants, and some applications of determinants.
- Key concepts covered include minors, cofactors, expanding determinants in terms of minors and cofactors, properties such as how determinants change with row/column operations, and using determinants to solve systems of linear equations.
- Examples are provided to demonstrate calculating determinants and using properties to simplify or prove identities about determinants.
BSC_COMPUTER _SCIENCE_UNIT-3_DISCRETE MATHEMATICSRai University
The document provides information about matrices including:
- Definitions of matrix, order of a matrix, types of matrices like row, column, square, null, identity, diagonal, scalar, and transpose matrices.
- Operations on matrices like addition, subtraction, scalar multiplication and matrix multiplication along with their properties.
- Examples of finding the sum of matrices.
- The document discusses various types of matrices like symmetric, skew-symmetric, singular, equal, negative, orthogonal, upper/lower triangular, trace, idempotent, involuntary, conjugate, unitary, Hermitian, skew-Hermitian matrices and minors of a matrix.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
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.
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 discusses matrices and their applications in solving systems of linear equations. It begins with defining what a matrix is and some special types of matrices including square, identity, zero, diagonal, commutative, triangular, and symmetric matrices. It also covers matrix operations like transpose and properties of determinants. The document then discusses using matrices to solve systems of linear equations through homogeneous systems, Gaussian elimination, Gaussian-Jordan elimination, and Cramer's rule. It includes examples and equations to illustrate the concepts and applications of matrices.
The rank of a matrix is the maximum number of linearly independent rows. A matrix has rank 0 only if it is the zero matrix. The inverse of a matrix A, denoted A^-1, is defined such that A*A^-1 = I, the identity matrix. To calculate the inverse, one takes the transpose of the cofactor matrix and divides each element by the determinant of the original matrix.
Matrices and its Applications to Solve Some Methods of Systems of Linear Equa...Rwan Kamal
This document is a research project submitted to the University of Duhok examining matrices and their applications in solving systems of linear equations. The project begins with an introduction to basic matrix concepts such as types of matrices, operations on matrices, and finding the inverse and determinant of matrices. It then discusses how matrices can be used to solve different types of systems of linear equations, including homogeneous systems, Gaussian elimination, Gaussian-Jordan elimination, and Cramer's rule. The document contains acknowledgments, contents, abstract, and introduction sections before presenting the technical matrix concepts and methods for solving systems of linear equations.
Matrices and its Applications to Solve Some Methods of Systems of Linear Equa...Abdullaا Hajy
This document is a research project submitted to the University of Duhok examining matrices and their applications in solving systems of linear equations. The project begins with an introduction to basic matrix concepts such as types of matrices, operations on matrices, and finding the inverse and determinant of matrices. It then discusses how matrices can be used to solve different types of systems of linear equations, including homogeneous systems, Gaussian elimination, Gaussian-Jordan elimination, and Cramer's rule. The document contains acknowledgments, contents, abstract, and introduction sections before presenting the technical matrix concepts and methods for solving systems of linear equations.
The document discusses matrices and their properties. It provides solutions to 10 problems involving finding orders and elements of matrices, constructing matrices based on given elements, solving systems of equations involving matrices, and determining the number of possible matrices of a given order and element values. Key details include: the possible orders of matrices with given numbers of elements, constructing matrices based on element definitions, solving for variables by equating matrix elements, and determining the number of 0/1 matrices of order 3x3 is 512.
The document defines matrices and provides examples of different types of matrices. It discusses key concepts such as rows, columns, dimensions, entries, addition, subtraction, and multiplication of matrices. It also covers special matrices like identity matrices, inverse matrices, transpose of matrices, and using matrices to solve systems of linear equations. The document is a comprehensive overview of matrices that defines fundamental terms and concepts.
The document discusses determinants and how to calculate them. It defines a determinant as a pure number associated with a square matrix. It provides an example of calculating the determinant of a 2x2 matrix. It then explains how to calculate minors and cofactors, which are used in Cramer's rule to solve systems of linear equations by calculating the determinants of the original and augmented matrices.
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.
For the following matrices, determine a cot of basis vectors for the.pdfeyebolloptics
For the following matrices, determine a cot of basis vectors for the null spaces the column
spaces. A = [3 2 -1 5] A = [2 1 -3 4 0 2 0 -2 8] Let P_n denote the set of all polynomial
functions of degree at most n Let a_o be a feed constant. Explain why H = [p(t) = a_0 + b_x| b
R} is not necessarily a vector subspace of P_x, Are there any values of a_o for which H will be a
subspace? If instead H = {p(t) = a + bt| a, b R}, (i.e., the constant term is allowed to vary over
all real numbers), show that H is a vector subspace of P_n.
Solution
Ans-
A matrix, in general sense, represents a
collection of information stored or arranged
in an orderly fashion. The mathematical
concept of a matrix refers to a set of numbers,
variables or functions ordered in rows and
columns. Such a set then can be defined as a
distinct entity, the matrix, and it can be
manipulated as a whole according to some
basic mathematical rules.
A matrix with 9 elements is shown below.
[
[[
[ ]
]]
]
=
==
=
aaa
=
==
=
253
A
131211
aaa
aaa
232221
333231
819
647
Matrix [A] has 3 rows and 3 columns. Each
element of matrix [A] can be referred to by its
row and column number. For example,
=
==
=a
23
6
A computer monitor with 800 horizontal
pixels and 600 vertical pixels can be viewed as
a matrix of 600 rows and 800 columns.
In order to create an image, each pixel is
filled with an appropriate colour.
ORDER OF A MATRIX
The order of a matrix is defined in terms of
its number of rows and columns.
Order of a matrix = No. of rows
×
××
×
No. of
columns
Matrix [A], therefore, is a matrix of order 3
×
××
×
3.
COLUMN MATRIX
A matrix with only one column is called a
column matrix or column vector.
ROW MATRIX
3
6
4
A matrix with only one row is called a row
matrix or row vector.
[
[[
[ ]
]]
]
653
SQUARE MATRIX
A matrix having the same number of rows
and columns is called a square matrix.
742
942
435
RECTANGULAR MATRIX
A matrix having unequal number of rows and
columns is called a rectangular matrix.
1735
13145
8292
REAL MATRIX
A matrix with all real elements is called a real
matrix
PRINCIPAL DIAGONAL and TRACE
OF A MATRIX
In a square matrix, the diagonal containing
the elements a
11
, a
22
, a
33
, a
44
, ……, a
is called
the principal or main diagonal.
The sum of all elements in the principal
diagonal is called the trace of the matrix.
The principal diagonal of the matrix
742
942
435
nn
is indicated by the dashed box. The trace of
the matrix is 2 + 3 + 9 = 14.
UNIT MATRIX
A square matrix in which all elements of the
principal diagonal are equal to 1 while all
other elements are zero is called the unit
matrix.
001
100
010
ZERO or NULL MATRIX
A matrix whose elements are all equal to zero
is called the null or zero matrix.
000
000
000
DIAGONAL MATRIX
If all elements except the elements of the
principal diagonal of a square matrix are
zero, the matrix is called a diagonal matrix.
002
900
030
RANK OF A MATRIX
The maximum number of linearly
independent rows of a matrix [A] is called
the rank of [A] and i.
The document discusses matrices and matrix operations. It defines a matrix as a rectangular array of elements arranged in rows and columns. It provides examples of matrix addition, multiplication, and properties such as commutativity and associativity of addition. Matrix multiplication is defined as the sum of the products of corresponding elements of the first matrix's rows and second matrix's columns. For multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second.
The document defines various types of matrices and matrix operations. It discusses the definitions of a matrix, types of matrices including rectangular, column, row, square, diagonal and unit matrices. It also defines transpose, symmetric, skew-symmetric and cofactor matrices. The document provides examples of calculating the minor and cofactor of matrix elements, as well as the adjoint and inverse of matrices using the formula for the inverse of a square matrix in terms of its determinant and adjoint.
This document provides an introduction to matrices. It defines a matrix as a rectangular array of numbers or other items arranged in rows and columns. Matrices are conventionally sized using the number of rows and columns. The document outlines basic matrix operations such as addition, subtraction, scalar multiplication, and matrix multiplication. It also defines key matrix types including identity, diagonal, triangular, and transpose matrices.
The document discusses matrices and determinants. It defines a matrix as a rectangular array of numbers arranged in rows and columns. A system of linear equations can be represented using matrices. Special types of matrices include diagonal matrices, where only the diagonal elements are non-zero, and identity matrices, where the diagonal elements are 1 and all others are 0. Matrices can be added or subtracted if they are the same size by adding or subtracting the corresponding elements. Determinants provide a value for a square matrix and are used in solutions to systems of linear equations.
The document defines and provides examples of different types of matrices such as square, diagonal, identity, and zero matrices. It also discusses matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. Key properties of these operations such as commutativity, associativity, and invertibility are covered. Matrix transpose and elementary row operations are also introduced.
1. The document discusses matrices, which are rectangular arrays of elements arranged in rows and columns. Common matrix operations include addition and multiplication.
2. Matrices come in several forms, including square, rectangular, null, diagonal, scalar, and identity matrices. Special types of square matrices include triangular superior and triangular inferior matrices.
3. For matrices to be equal, the elements in each corresponding position must be equal. Matrix operations like addition involve performing the operation on corresponding elements.
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.
1. The document discusses matrices and various matrix operations.
2. It defines what a matrix is and different types of matrices such as row matrix, column matrix, square matrix, diagonal matrix, triangular matrix, and null matrix.
3. It covers matrix addition and multiplication. Matrix addition involves adding corresponding elements of two matrices of the same size. Matrix multiplication is only defined if the number of columns of the first matrix equals the number of rows of the second matrix.
This document discusses applications of derivatives, including determining rates of change, finding equations of tangents and normals to curves, locating maxima and minima, and determining intervals where functions are increasing or decreasing. It provides examples of using derivatives to find rates of change for various quantities like area of a circle, volume of a cube, and enclosed area of a circular wave. It also introduces the concept of marginal cost and revenue. Finally, it discusses using the sign of the derivative to determine if a function is increasing, decreasing, or constant on an interval using the first derivative test.
This document provides an introduction to the concepts of continuity and differentiability in calculus. It begins by giving two informal examples of functions that are and aren't continuous at a point to build intuition. It then provides a formal definition of continuity as the limit of a function at a point equaling the function value at that point. Several examples are worked through to demonstrate checking continuity at points and for entire functions. The document introduces the concept of limits approaching infinity to discuss the continuity of functions like 1/x. Overall, it lays the groundwork for understanding continuity and differentiability through examples and definitions.
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.
The document discusses inverse trigonometric functions. It defines the inverse functions of sine, cosine, tangent, cotangent, secant and cosecant by restricting their domains such that they become one-to-one functions. The principal value branches are defined for each inverse function, with their domains and ranges specified. Graphs are provided to illustrate the inverse relationships between the trigonometric functions and their inverses. Properties like how to obtain the graph of an inverse function from the original are also outlined.
(1) The document discusses relations and functions in mathematics. It defines different types of relations such as empty relation, universal relation, equivalence relation, reflexive relation, symmetric relation and transitive relation. (2) It provides examples to illustrate these relations and checks whether given relations satisfy the properties. (3) The document also discusses that an equivalence relation partitions a set into mutually exclusive equivalence classes.
This document is the foreword and preface for a mathematics textbook for Class XII in India. It discusses the development of the textbook based on the National Curriculum Framework to link classroom learning with students' lives outside of school. The textbook aims to discourage rote learning and encourage student-centered learning. It provides concise summaries of each chapter and incorporates examples, exercises, and historical context to motivate learning. The foreword acknowledges the many teachers and experts who contributed to developing the textbook.
This document contains exercises related to mathematics including:
1) Questions about relations being reflexive, symmetric, and transitive.
2) Matrix addition, multiplication, and inverse operations.
3) Questions involving functions, their compositions and inverses.
4) Questions testing knowledge of properties like commutativity and associativity.
The document provides problems and answers related to foundational mathematics concepts.
This document discusses various methods of proving mathematical propositions, including direct proof, indirect proof, proof by contradiction, proof by cases, and proof by mathematical induction. It provides examples to illustrate each method. Direct proof involves directly deducing the conclusion from the given statements, while indirect proof establishes an equivalent proposition. Proof by contradiction assumes the negation of the statement to be proved and arrives at a contradiction. Proof by cases considers all possible cases of the hypothesis. Mathematical induction proves a statement for all natural numbers based on proving it for the base case and assuming it is true for some arbitrary case k.
This document discusses mathematical modeling and provides several examples to illustrate the process. It begins by defining mathematical modeling as using mathematical concepts to describe real-world phenomena. It then lists several examples of problems that can be solved using mathematical modeling. The document goes on to outline the basic principles and steps of mathematical modeling, including identifying the problem, relevant variables and data, assumptions, governing equations, and comparing the model's results to observations. Finally, it provides several examples walking through applying these steps to problems like estimating raw material needs for a business, maximizing production output based on constraints, and more.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
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.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.