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 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.
This document discusses real vector spaces and provides examples of determining whether a set with defined operations is a vector space. Some key points covered include:
- The definition of a vector space and properties it must satisfy, such as closure under addition and scalar multiplication.
- Examples of determining if a set is a vector space by checking if it satisfies the necessary properties.
- The definition of a subspace, and using properties of closure under operations to determine if a subset is a subspace.
- The concept of a linear combination of vectors and using an augmented matrix to determine if a vector can be written as a linear combination of other vectors.
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
This document discusses key concepts in quantum mechanics including wave functions, operators, linear vector spaces, inner products, orthogonal and orthonormal bases, Hilbert spaces, and the expansion theorem. It defines wave functions and operators as the two main constructs in quantum mechanics. It also explains that the natural language of quantum mechanics is linear algebra and describes concepts like linear vector spaces, inner products, orthogonal and orthonormal bases, and Hilbert spaces in the context of quantum mechanics.
This document discusses vector spaces and subspaces. It begins by defining a vector space as a set V with two operations, vector addition and scalar multiplication, that satisfy certain properties. Examples of vector spaces include R2 and the space of real polynomials of degree n or less.
It then defines a subspace as a subset of a vector space that is itself a vector space under the inherited operations. For a subset to be a subspace, it must be closed under vector addition and scalar multiplication, and contain the zero vector. Examples given include lines and planes through the origin in R3.
The span of a set S of vectors is defined as the set of all linear combinations of the vectors in S, and it
This document provides information about vector spaces and subspaces. It defines a vector space as a set of objects called vectors that can be added together and multiplied by scalars, subject to certain rules. A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. The null space of a matrix is the set of solutions to the homogeneous equation Ax=0 and is a subspace. The column space of a matrix is the set of all linear combinations of its columns and is also a subspace. Examples are provided to illustrate these concepts.
The document discusses vector spaces and related linear algebra concepts. It defines vector spaces and lists the axioms that must be satisfied. Examples of vector spaces include the set of all pairs of real numbers and the space of 2x2 symmetric matrices. The document also discusses subspaces, linear combinations, span, basis, dimension, row space, column space, null space, rank, nullity, and change of basis. It provides examples and explanations of these fundamental linear algebra topics.
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
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.
This document discusses real vector spaces and provides examples of determining whether a set with defined operations is a vector space. Some key points covered include:
- The definition of a vector space and properties it must satisfy, such as closure under addition and scalar multiplication.
- Examples of determining if a set is a vector space by checking if it satisfies the necessary properties.
- The definition of a subspace, and using properties of closure under operations to determine if a subset is a subspace.
- The concept of a linear combination of vectors and using an augmented matrix to determine if a vector can be written as a linear combination of other vectors.
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
This document discusses key concepts in quantum mechanics including wave functions, operators, linear vector spaces, inner products, orthogonal and orthonormal bases, Hilbert spaces, and the expansion theorem. It defines wave functions and operators as the two main constructs in quantum mechanics. It also explains that the natural language of quantum mechanics is linear algebra and describes concepts like linear vector spaces, inner products, orthogonal and orthonormal bases, and Hilbert spaces in the context of quantum mechanics.
This document discusses vector spaces and subspaces. It begins by defining a vector space as a set V with two operations, vector addition and scalar multiplication, that satisfy certain properties. Examples of vector spaces include R2 and the space of real polynomials of degree n or less.
It then defines a subspace as a subset of a vector space that is itself a vector space under the inherited operations. For a subset to be a subspace, it must be closed under vector addition and scalar multiplication, and contain the zero vector. Examples given include lines and planes through the origin in R3.
The span of a set S of vectors is defined as the set of all linear combinations of the vectors in S, and it
This document provides information about vector spaces and subspaces. It defines a vector space as a set of objects called vectors that can be added together and multiplied by scalars, subject to certain rules. A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. The null space of a matrix is the set of solutions to the homogeneous equation Ax=0 and is a subspace. The column space of a matrix is the set of all linear combinations of its columns and is also a subspace. Examples are provided to illustrate these concepts.
The document discusses vector spaces and related linear algebra concepts. It defines vector spaces and lists the axioms that must be satisfied. Examples of vector spaces include the set of all pairs of real numbers and the space of 2x2 symmetric matrices. The document also discusses subspaces, linear combinations, span, basis, dimension, row space, column space, null space, rank, nullity, and change of basis. It provides examples and explanations of these fundamental linear algebra topics.
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
The document provides an overview of vector spaces and related linear algebra concepts. It defines vector spaces, subspaces, basis, dimension, and rank. Key points include:
- A vector space is a set that is closed under vector addition and scalar multiplication. It must satisfy certain axioms.
- A subspace is a subset of a vector space that is also a vector space.
- A basis is a minimal set of linearly independent vectors that span the entire vector space. The dimension of a vector space is the number of vectors in its basis.
- The rank of a matrix is the number of linearly independent rows in its row-reduced echelon form. It provides a measure of the matrix's linear
This document provides notes on vector spaces, which are fundamental objects in linear algebra. It begins with examples of vector spaces such as R2, R3, C2, C3 and defines vector spaces more generally as sets that are closed under vector addition and scalar multiplication and satisfy other properties like the existence of additive identities. It then provides several examples of vector spaces including the set of all n-tuples over a field, the set of all m×n matrices, the set of differentiable functions on an interval, and the set of polynomials with coefficients in a field.
The document defines a subspace as a non-empty subset W of a vector space V that is itself a vector space under the operations defined on V. It notes that every vector space has at least two subspaces: itself and the zero subspace containing only the zero vector. To prove that W is a subspace of V, we only need to verify that W is closed under the vector space operations. Examples are provided to illustrate this, such as showing that the set W={(x,0,0)| x in R} is a subspace of R3 by verifying it is closed under vector addition and scalar multiplication.
The document provides information about a linear algebra and vector calculus assignment for mechanical engineering students at L.D. College of Engineering in Ahmedabad, India. It includes the names of 10 students, an outline of 8 topics to be covered, and sample definitions, examples, and explanations related to those topics, such as definitions of vector spaces and subspaces, linear combinations, linear independence, and span of a set of vectors.
The document defines key concepts in vector spaces including vector space, subspace, span of a set of vectors, and basis. It provides examples to illustrate these concepts. Specifically:
- A vector space is a set of objects called vectors that can be added together and multiplied by scalars, satisfying certain properties.
- A subspace is a subset of a vector space that is itself a vector space under the operations of the original space.
- The span of a set of vectors S is the set of all possible linear combinations of the vectors in S.
- A basis is a set of vectors that spans a vector space and is linearly independent. It provides a standard representation for vectors in the space.
Chapter 4: Vector Spaces - Part 1/Slides By PearsonChaimae Baroudi
This document defines vectors and vector spaces. It begins by defining vectors in 2D and 3D space as matrices and describes operations like addition, scalar multiplication, and subtraction. It then defines a vector space as a set of vectors that satisfies 10 axioms related to these operations. Examples of vector spaces include the set of 2D and 3D vectors, sets of matrices, and sets of polynomials. The document also defines subspaces and proves that the span of a set of vectors in a vector space forms a subspace.
The document discusses vector spaces and related concepts:
1) It defines a vector space as a set V with vector addition and scalar multiplication operations that satisfy certain properties. Examples of vector spaces include R2, the plane in R3, and the space of real polynomials.
2) A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication and thus forms a vector space with the inherited operations. Examples given include the x-axis in Rn and solution spaces of linear differential equations.
3) The span of a set of vectors is the smallest subspace that contains those vectors, consisting of all possible linear combinations of the vectors in the set.
The document provides examples to illustrate how to find the eigenvalues and eigenvectors of a matrix.
1) For a 2x2 matrix, the characteristic polynomial is computed by taking the determinant of the matrix minus the identity matrix. The roots of the characteristic polynomial are the eigenvalues. The corresponding eigenvectors are found by solving the original eigenvalue equation.
2) For a triangular matrix, the eigenvalues are the diagonal elements. The eigenvectors are found by setting rows corresponding to non-diagonal elements to zero.
3) The document provides a numerical example to demonstrate finding the eigenvalues (3, 1, -2) and eigenvectors of a 3x3 matrix.
The document defines linear independence and dependence of vectors and discusses some key properties:
- A set of vectors is linearly independent if the only solution to their linear combination equaling the zero vector is the trivial solution with all coefficients equal to 0.
- A set is linearly dependent if at least one vector can be written as a linear combination of the others.
- A set of one vector is independent if it is not the zero vector. A set of two vectors is dependent if one is a multiple of the other.
- If a set contains more vectors than the dimension of the vector space, the set must be dependent since there are more variables than equations.
Chapter 4: Vector Spaces - Part 3/Slides By PearsonChaimae Baroudi
1. A basis for a vector space is a set of linearly independent vectors that span the entire space. The standard basis for Rn is the set of n unit vectors e1, e2, ..., en.
2. The dimension of a vector space is the number of vectors in any of its bases. A vector space with dimension n has bases that contain exactly n vectors.
3. The null space of a matrix A consists of all vectors x such that Ax = 0. The dimension of the null space is called the nullity of A, and the null space always has a basis.
The document discusses eigenvalues and eigenvectors of linear transformations and matrices. It begins by defining a diagonalizable matrix as one that can be transformed into a diagonal matrix through a change of basis. It then defines eigenvalues and eigenvectors for both linear transformations and matrices. The characteristic polynomial of a matrix is introduced, which has roots that are the eigenvalues of the matrix. It is shown that the algebraic multiplicity of an eigenvalue is equal to its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace. The algebraic multiplicity is always greater than or equal to the geometric multiplicity.
1) An inner product space is a vector space with an inner product defined that satisfies certain properties like linearity and positive-definiteness.
2) The Gram-Schmidt process is used to transform a basis into an orthogonal basis and then an orthonormal basis by successively subtracting projections.
3) The angle between two vectors in an inner product space can be computed using the inner product and the norms of the vectors.
Chapter 4: Vector Spaces - Part 2/Slides By PearsonChaimae Baroudi
This document discusses linear combinations and independence of vectors. It defines linear combinations as vectors that can be expressed as a sum of other vectors with scalar coefficients. A set of vectors is linearly dependent if one vector can be written as a linear combination of the others, and linearly independent otherwise. The span of a set of vectors is the set of all their linear combinations, and spans the entire space if and only if the vectors are independent. The null space of a matrix contains vectors that solve the homogeneous equation Ax=0. Examples demonstrate determining if sets of vectors are linearly dependent or independent.
The document provides an introduction to linear algebra concepts for machine learning. It defines vectors as ordered tuples of numbers that express magnitude and direction. Vector spaces are sets that contain all linear combinations of vectors. Linear independence and basis of vector spaces are discussed. Norms measure the magnitude of a vector, with examples given of the 1-norm and 2-norm. Inner products measure the correlation between vectors. Matrices can represent linear operators between vector spaces. Key linear algebra concepts such as trace, determinant, and matrix decompositions are outlined for machine learning applications.
This document provides an overview of row space, column space, and null space of matrices. It defines these concepts and gives examples of finding bases for the row space, column space, and null space. It also introduces the rank-nullity theorem and defines the rank and nullity of a matrix. Examples are provided to demonstrate calculating the rank and nullity. The document appears to be teaching notes for a linear algebra course.
This document discusses vector spaces and related concepts such as subspaces, linear combinations, linear independence, spanning sets, bases, and dimension. It begins by defining a vector space and providing examples. It then covers subspaces and shows that every vector space has at least two subspaces: the zero vector space and the entire vector space. The document also discusses linear combinations, linear independence, spanning sets, bases, and notes some key properties such as the uniqueness of the basis representation in a vector space.
The document discusses convergence of sequences and power series. It defines convergence of a sequence and states that the limit of a convergent sequence is unique. It also discusses Taylor series and Laurent series, stating that if a function f(z) is analytic inside a circle C with center z0, its Taylor series representation about z0 will converge to f(z) for all z inside C. Similarly, if f(z) is analytic in an annular region bounded by two concentric circles, its Laurent series will represent f(z) in that region.
In general, we can find the coordinates of a vector u with respect to a given basis B by solving ABuB = u, for uB, where ABis the matrix whose columns are the vectors in B. ABis called thechange of basis matrix for B.
This document discusses linear transformations and their properties. It defines a linear transformation as a function between vector spaces that preserves vector addition and scalar multiplication. The kernel of a linear transformation is the set of vectors mapped to the zero vector, and is a subspace of the domain. The range is the set of images of all vectors under the transformation. Matrices can represent linear transformations, with the matrix equation representing the transformation of vectors. Examples are provided to illustrate key concepts such as kernels, ranges, and matrix representations of linear transformations.
The document provides an overview of vector spaces and related linear algebra concepts. It defines vector spaces, subspaces, basis, dimension, and rank. Key points include:
- A vector space is a set that is closed under vector addition and scalar multiplication. It must satisfy certain axioms.
- A subspace is a subset of a vector space that is also a vector space.
- A basis is a minimal set of linearly independent vectors that span the entire vector space. The dimension of a vector space is the number of vectors in its basis.
- The rank of a matrix is the number of linearly independent rows in its row-reduced echelon form. It provides a measure of the matrix's linear
This document provides notes on vector spaces, which are fundamental objects in linear algebra. It begins with examples of vector spaces such as R2, R3, C2, C3 and defines vector spaces more generally as sets that are closed under vector addition and scalar multiplication and satisfy other properties like the existence of additive identities. It then provides several examples of vector spaces including the set of all n-tuples over a field, the set of all m×n matrices, the set of differentiable functions on an interval, and the set of polynomials with coefficients in a field.
The document defines a subspace as a non-empty subset W of a vector space V that is itself a vector space under the operations defined on V. It notes that every vector space has at least two subspaces: itself and the zero subspace containing only the zero vector. To prove that W is a subspace of V, we only need to verify that W is closed under the vector space operations. Examples are provided to illustrate this, such as showing that the set W={(x,0,0)| x in R} is a subspace of R3 by verifying it is closed under vector addition and scalar multiplication.
The document provides information about a linear algebra and vector calculus assignment for mechanical engineering students at L.D. College of Engineering in Ahmedabad, India. It includes the names of 10 students, an outline of 8 topics to be covered, and sample definitions, examples, and explanations related to those topics, such as definitions of vector spaces and subspaces, linear combinations, linear independence, and span of a set of vectors.
The document defines key concepts in vector spaces including vector space, subspace, span of a set of vectors, and basis. It provides examples to illustrate these concepts. Specifically:
- A vector space is a set of objects called vectors that can be added together and multiplied by scalars, satisfying certain properties.
- A subspace is a subset of a vector space that is itself a vector space under the operations of the original space.
- The span of a set of vectors S is the set of all possible linear combinations of the vectors in S.
- A basis is a set of vectors that spans a vector space and is linearly independent. It provides a standard representation for vectors in the space.
Chapter 4: Vector Spaces - Part 1/Slides By PearsonChaimae Baroudi
This document defines vectors and vector spaces. It begins by defining vectors in 2D and 3D space as matrices and describes operations like addition, scalar multiplication, and subtraction. It then defines a vector space as a set of vectors that satisfies 10 axioms related to these operations. Examples of vector spaces include the set of 2D and 3D vectors, sets of matrices, and sets of polynomials. The document also defines subspaces and proves that the span of a set of vectors in a vector space forms a subspace.
The document discusses vector spaces and related concepts:
1) It defines a vector space as a set V with vector addition and scalar multiplication operations that satisfy certain properties. Examples of vector spaces include R2, the plane in R3, and the space of real polynomials.
2) A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication and thus forms a vector space with the inherited operations. Examples given include the x-axis in Rn and solution spaces of linear differential equations.
3) The span of a set of vectors is the smallest subspace that contains those vectors, consisting of all possible linear combinations of the vectors in the set.
The document provides examples to illustrate how to find the eigenvalues and eigenvectors of a matrix.
1) For a 2x2 matrix, the characteristic polynomial is computed by taking the determinant of the matrix minus the identity matrix. The roots of the characteristic polynomial are the eigenvalues. The corresponding eigenvectors are found by solving the original eigenvalue equation.
2) For a triangular matrix, the eigenvalues are the diagonal elements. The eigenvectors are found by setting rows corresponding to non-diagonal elements to zero.
3) The document provides a numerical example to demonstrate finding the eigenvalues (3, 1, -2) and eigenvectors of a 3x3 matrix.
The document defines linear independence and dependence of vectors and discusses some key properties:
- A set of vectors is linearly independent if the only solution to their linear combination equaling the zero vector is the trivial solution with all coefficients equal to 0.
- A set is linearly dependent if at least one vector can be written as a linear combination of the others.
- A set of one vector is independent if it is not the zero vector. A set of two vectors is dependent if one is a multiple of the other.
- If a set contains more vectors than the dimension of the vector space, the set must be dependent since there are more variables than equations.
Chapter 4: Vector Spaces - Part 3/Slides By PearsonChaimae Baroudi
1. A basis for a vector space is a set of linearly independent vectors that span the entire space. The standard basis for Rn is the set of n unit vectors e1, e2, ..., en.
2. The dimension of a vector space is the number of vectors in any of its bases. A vector space with dimension n has bases that contain exactly n vectors.
3. The null space of a matrix A consists of all vectors x such that Ax = 0. The dimension of the null space is called the nullity of A, and the null space always has a basis.
The document discusses eigenvalues and eigenvectors of linear transformations and matrices. It begins by defining a diagonalizable matrix as one that can be transformed into a diagonal matrix through a change of basis. It then defines eigenvalues and eigenvectors for both linear transformations and matrices. The characteristic polynomial of a matrix is introduced, which has roots that are the eigenvalues of the matrix. It is shown that the algebraic multiplicity of an eigenvalue is equal to its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace. The algebraic multiplicity is always greater than or equal to the geometric multiplicity.
1) An inner product space is a vector space with an inner product defined that satisfies certain properties like linearity and positive-definiteness.
2) The Gram-Schmidt process is used to transform a basis into an orthogonal basis and then an orthonormal basis by successively subtracting projections.
3) The angle between two vectors in an inner product space can be computed using the inner product and the norms of the vectors.
Chapter 4: Vector Spaces - Part 2/Slides By PearsonChaimae Baroudi
This document discusses linear combinations and independence of vectors. It defines linear combinations as vectors that can be expressed as a sum of other vectors with scalar coefficients. A set of vectors is linearly dependent if one vector can be written as a linear combination of the others, and linearly independent otherwise. The span of a set of vectors is the set of all their linear combinations, and spans the entire space if and only if the vectors are independent. The null space of a matrix contains vectors that solve the homogeneous equation Ax=0. Examples demonstrate determining if sets of vectors are linearly dependent or independent.
The document provides an introduction to linear algebra concepts for machine learning. It defines vectors as ordered tuples of numbers that express magnitude and direction. Vector spaces are sets that contain all linear combinations of vectors. Linear independence and basis of vector spaces are discussed. Norms measure the magnitude of a vector, with examples given of the 1-norm and 2-norm. Inner products measure the correlation between vectors. Matrices can represent linear operators between vector spaces. Key linear algebra concepts such as trace, determinant, and matrix decompositions are outlined for machine learning applications.
This document provides an overview of row space, column space, and null space of matrices. It defines these concepts and gives examples of finding bases for the row space, column space, and null space. It also introduces the rank-nullity theorem and defines the rank and nullity of a matrix. Examples are provided to demonstrate calculating the rank and nullity. The document appears to be teaching notes for a linear algebra course.
This document discusses vector spaces and related concepts such as subspaces, linear combinations, linear independence, spanning sets, bases, and dimension. It begins by defining a vector space and providing examples. It then covers subspaces and shows that every vector space has at least two subspaces: the zero vector space and the entire vector space. The document also discusses linear combinations, linear independence, spanning sets, bases, and notes some key properties such as the uniqueness of the basis representation in a vector space.
The document discusses convergence of sequences and power series. It defines convergence of a sequence and states that the limit of a convergent sequence is unique. It also discusses Taylor series and Laurent series, stating that if a function f(z) is analytic inside a circle C with center z0, its Taylor series representation about z0 will converge to f(z) for all z inside C. Similarly, if f(z) is analytic in an annular region bounded by two concentric circles, its Laurent series will represent f(z) in that region.
In general, we can find the coordinates of a vector u with respect to a given basis B by solving ABuB = u, for uB, where ABis the matrix whose columns are the vectors in B. ABis called thechange of basis matrix for B.
This document discusses linear transformations and their properties. It defines a linear transformation as a function between vector spaces that preserves vector addition and scalar multiplication. The kernel of a linear transformation is the set of vectors mapped to the zero vector, and is a subspace of the domain. The range is the set of images of all vectors under the transformation. Matrices can represent linear transformations, with the matrix equation representing the transformation of vectors. Examples are provided to illustrate key concepts such as kernels, ranges, and matrix representations of linear transformations.
This document discusses linear transformations between vector spaces. It begins by defining a linear transformation as a function between vector spaces that satisfies the properties of vector addition and scalar multiplication. It then provides examples of standard linear transformations like the matrix transformation and zero transformation. The document also covers properties of linear transformations such as how they are determined by the images of basis vectors. Finally, it provides applications of linear operators like reflection, rotation, and shear transformations.
The document discusses linear transformations and mathematical methods. It provides the syllabus for a course covering topics like matrices, eigenvalues and eigenvectors, linear transformations, solutions to non-linear systems, curve fitting, numerical integration, Fourier series, and partial differential equations. It specifically covers properties of eigenvalues and eigenvectors, the Cayley-Hamilton theorem, diagonalizing matrices, and modal and spectral matrices in unit III on linear transformations.
The document defines and provides examples of linear transformations. It then presents a question asking to find the matrix of a linear transformation between two vector spaces. The solution shows that:
1) The vector spaces have standard bases of {[1,0,0],[0,1,0],[0,0,1]} and {[1,0],[0,1]}.
2) The matrix of the linear transformation is A = [[3,4,9],[5,3,2]].
3) For a given linear transformation T defined by a matrix A, the transformation can be expressed in terms of coordinates as T(x)=A*x.
The document discusses using linear algebra to solve chemistry problems involving balancing chemical equations and determining volumes of chemical solutions. It provides an example of using a system of equations represented in matrix form to determine that the volumes of solutions A, B, and C are 1.5 cm3, 3.1 cm3, and 2.2 cm3 based on the amounts of chemical produced under different concentration conditions. The document also demonstrates how to use the law of conservation of matter and a similar technique to balance the chemical equation C2H6 + O2 → CO2 + H2O.
This document provides an introduction to linear transformations. It defines a linear transformation as a function that maps one vector space to another while preserving vector addition and scalar multiplication. Key concepts discussed include the domain, co-domain, range, and pre-image of a linear transformation. Examples are given to demonstrate linear transformations and functions that are not linear transformations. The relationship between linear transformations and matrices is also explained.
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.
This document discusses various applications of linear algebra in different fields such as abstract thinking, chemistry, coding theory, cryptography, economics, elimination theory, games, genetics, geometry, graph theory, heat distribution, image compression, linear programming, Markov chains, networking, and sociology. It provides examples of how linear algebra concepts such as systems of linear equations and matrix operations are used in topics like balancing chemical equations, error detection in coding, encryption/decryption, economic models, genetic inheritance, and finding lines and circles in geometry.
- 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.
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.
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.
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.
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.
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.
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.
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.
- 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 discusses various data transformations that can be used to satisfy assumptions of normality, homogeneity of variance, and linearity when analyzing metric variables. It describes how to compute logarithmic, square root, inverse, and square transformations in SPSS. Adjustments may need to be added to the values when computing the transformations depending on whether the original variable is positively or negatively skewed. The transformed variables are added as new variables to the SPSS data file.
Graphs are used in physics to show relationships between variables. A linear graph indicates a direct proportional relationship between variables. The slope of a linear graph is calculated by taking the rise over the run and represents the ratio of change in the dependent variable to the change in the independent variable. For nonlinear relationships, manipulating the variables, such as squaring or taking reciprocals, can linearize the relationship. The slope of the resulting linear graph then represents a physical quantity defined by the original equation.
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).
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
First part of description of Matrix Calculus at Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations please visit my website at
http://www.solohermelin.com.
I am Fabian H. I am a Calculus Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics, Deakin University, Australia. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
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Special second order non symmetric fitted method for singularAlexander Decker
The document presents a special second order non-symmetric fitted difference method for solving singularly perturbed two-point boundary value problems with boundary layers. The method introduces a fitting factor in the finite difference scheme to account for rapid changes in the boundary layer. The fitting factor is derived from singular perturbation theory. The method is applied to problems with left-end and right-end boundary layers. A tridiagonal system is obtained and solved using the discrete invariant embedding algorithm. Numerical results illustrate the accuracy of the proposed method.
Liquid-liquid extraction is a separation technique used when components have different solubilities in two immiscible liquids, rather than different volatilities. It involves mixing the liquids to contact the components, then separating them. There are two main types of systems - those forming one pair of partially miscible liquids, shown on a closed ternary diagram, and those forming two pairs, shown on an open ternary diagram. Key aspects of a stage include the contact and separation of solvent and feed mixtures, and definitions of extract, raffinate, mass flow rates and compositions. Problems apply concepts like equilibrium lines, operating lines, and calculations for number of stages and compositions.
The document discusses liquid-liquid extraction as a separation technique. It provides three main points:
1. Liquid-liquid extraction separates species according to differences in solubility between two immiscible liquids, rather than volatility as in distillation.
2. The key components in liquid-liquid extraction are a solute, solvent, and inert components that form two separate liquid phases - an extract and raffinate.
3. There are two main types of systems - one forming one pair of partially miscible liquids leading to a closed ternary diagram, and one forming two pairs forming an open ternary diagram.
The document then discusses various aspects of setting up counter-current multi-
1. The document discusses functions of several variables and partial differentiation.
2. It provides examples of functions with different domains and ranges, such as f(x,y) = x^2 + y^2 which has a domain of all real numbers and a range of non-negative real numbers.
3. It also examines how changing variables in functions impacts the output, like how increasing humidity by 20% at 80 degrees Fahrenheit increases the heat index by about 3 units.
The document contains examples of functions of several variables and their domains and ranges. It provides equations for various functions and graphs their surfaces over different domains. Some key examples include functions defined by equations like x2 + y2 = 1, 2, 3 and functions where increasing one variable by a fixed amount increases the output by a fixed amount.
The document discusses vector spaces and their properties. It defines a vector space as a collection of vectors that can be added and scaled by real numbers, while satisfying certain properties like closure and distributivity. Examples of vector spaces include Rn, the space of matrices, and function spaces. A subspace is a subset of a vector space that is also a vector space. The column space of a matrix contains all linear combinations of its columns and is an important subspace.
The document defines vectors and describes various vector operations that can be performed, including:
- Graphical vector addition using the tip-to-tail method
- Numerical representation of vectors using magnitude, direction, and components
- Resolution of a vector into perpendicular components
- Composition and decomposition of vectors using graphical and numerical methods
- Scalar multiplication and subtraction of vectors
It also provides examples of how to use vectors to solve problems graphically or numerically.
The document discusses key concepts in vectors including:
- Vectors can be represented geometrically as arrows or algebraically as ordered lists of numbers in a coordinate system.
- The two fundamental vector operations are vector addition and scalar multiplication. Vector addition involves combining the components of vectors, while scalar multiplication scales the magnitude and direction of a vector.
- Basis vectors define a coordinate system. Any vector can be written as a linear combination of basis vectors using scalar multiplication and vector addition. In 2D, the standard basis vectors are i and j along the x- and y-axes.
- The linear span of vectors is the set of all possible linear combinations of those vectors. If the vectors are linearly independent
The document discusses vector spaces and related concepts. It begins by defining a vector space as a non-empty set V with defined operations of vector addition and scalar multiplication that satisfy certain axioms. Examples of vector spaces include Rn and the set of m×n matrices. A subspace is a subset of a vector space that is also a vector space under the defined operations. Properties of subspaces and examples are provided. Linear combinations, linear independence, spanning sets, and the span of a set of vectors are then defined and explained.
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.
I am Manuela B. I am a Linear Algebra Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick. I have been helping students with their assignments for the past 9 years. I solve assignments related to Linear Algebra.
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You can also call on +1 678 648 4277 for any assistance with Linear Algebra Assignment.
This document provides an overview of concepts from linear algebra that are necessary for understanding quantum mechanics. It reviews vectors, vector spaces, linear independence, bases, linear operators, and complex numbers. It then introduces key concepts for quantum mechanics, including Dirac notation, inner products, outer products, eigenvalues and eigenvectors, unitary and Hermitian operators, and tensor products. The goal is to cover the necessary mathematical foundations and notations systematically to enable the study of quantum mechanics postulates.
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.
Vector space interpretation_of_random_variablesGopi Saiteja
This document discusses vector space interpretation of random variables. It begins by introducing vector spaces and their properties such as closure under addition and scalar multiplication. Random variables can be interpreted as elements of a vector space. Inner products, norms, orthogonality and projections are discussed in the context of both vector spaces and random variables. Interpreting expectations as inner products allows treating random variables as vectors in an inner product space.
This document defines key concepts in linear algebra including vector spaces, vectors, and operations on vectors such as addition and scalar multiplication. It specifically focuses on the vector space Rn:
- Rn is the set of all n-tuples of real numbers, which forms a vector space under component-wise addition and scalar multiplication.
- The dot product and norm are defined for vectors in Rn and used to determine properties like orthogonality and the angle between vectors.
- Examples show how to perform vector operations in Rn like addition, scalar multiplication, finding the dot product and norm, and determining if vectors are orthogonal.
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. 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 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.
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.
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
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
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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.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
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.
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.
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.
1. Announcements
Quiz 1 after lecture.
Today (Jan 20, Wed) is the last day to drop this class with no
academic penalty (No record on transcript). No refunds.
2 Corrections made to yesterday's slide (change 20 to 16 and
R3-R2 to R3-R1)
2. Last Class...
Suppose we have a set of vectors v1 . . . vp in Rn . If the vector
equation x1 v1 + . . . + xp vp = 0 has ONLY THE TRIVIAL SOLUTION
we say that the set of vectors is linearly independent.
3. Last Class...
Suppose we have a set of vectors v1 . . . vp in Rn . If the vector
equation x1 v1 + . . . + xp vp = 0 has ONLY THE TRIVIAL SOLUTION
we say that the set of vectors is linearly independent.
In other words, if we can nd atleast ONE non-zero weight c1 , c2 ,
. . ., cp such that c1 v1 + . . . + cp vp = 0, then the set v1 . . . vp is a
LINEARLY DEPENDENT set
4. Linear Independence Check by Inspection
Some set of vectors have some obvious properties that will help you
decide whether they are linearly independent without doing any row
operations.
5. Linear Independence Check by Inspection
Some set of vectors have some obvious properties that will help you
decide whether they are linearly independent without doing any row
operations.
If you have only one vector v and if it is not the zero vector
then v is linearly independent. (x1 v = 0 has only trivial
solution)
6. Linear Independence Check by Inspection
Some set of vectors have some obvious properties that will help you
decide whether they are linearly independent without doing any row
operations.
If you have only one vector v and if it is not the zero vector
then v is linearly independent. (x1 v = 0 has only trivial
solution)
The zero vector is linearly dependent. (The equation x1 0 = 0
has many solutions.
7. Linear Independence Check by Inspection
Some set of vectors have some obvious properties that will help you
decide whether they are linearly independent without doing any row
operations.
If you have only one vector v and if it is not the zero vector
then v is linearly independent. (x1 v = 0 has only trivial
solution)
The zero vector is linearly dependent. (The equation x1 0 = 0
has many solutions.
If two vectors are given and one vector is a multiple of the
other then the two vectors are LINEARLY DEPENDENT
8. Linear Independence Check by Inspection
Some set of vectors have some obvious properties that will help you
decide whether they are linearly independent without doing any row
operations.
If you have only one vector v and if it is not the zero vector
then v is linearly independent. (x1 v = 0 has only trivial
solution)
The zero vector is linearly dependent. (The equation x1 0 = 0
has many solutions.
If two vectors are given and one vector is a multiple of the
other then the two vectors are LINEARLY DEPENDENT
If a set has more vectors than there are entries in each vector,
then the set is LINEARLY DEPENDENT.
9. Linear Independence Check by Inspection
Some set of vectors have some obvious properties that will help you
decide whether they are linearly independent without doing any row
operations.
If you have only one vector v and if it is not the zero vector
then v is linearly independent. (x1 v = 0 has only trivial
solution)
The zero vector is linearly dependent. (The equation x1 0 = 0
has many solutions.
If two vectors are given and one vector is a multiple of the
other then the two vectors are LINEARLY DEPENDENT
If a set has more vectors than there are entries in each vector,
then the set is LINEARLY DEPENDENT.
If a set of vectors contains the zero vector, the set is linearly
dependent.
10. Examples
Problem 16 section 1.7 Determine by inspection whether the
vectors are linearly independent. Justify your answer.
4 6
−2 , −3
6 9
11. Examples
Problem 16 section 1.7 Determine by inspection whether the
vectors are linearly independent. Justify your answer.
4 6
−2 , −3
6 9
Solution The second vector is 1.5 times the rst vector. So these
vectors are linearly dependent.
12. Examples
Problem 18 section 1.7 Determine by inspection whether the
vectors are linearly independent. Justify your answer.
4 −1 2 8
, , ,
4 3 5 1
13. Examples
Problem 18 section 1.7 Determine by inspection whether the
vectors are linearly independent. Justify your answer.
4 −1 2 8
, , ,
4 3 5 1
Solution There are 4 vectors and each vector has 2 entries each. So
this set is linearly dependent.
14. Examples
Problem 20 section 1.7 Determine by inspection whether the
vectors are linearly independent. Justify your answer.
1 −2 0
4 , 5 , 0
−7 3 0
15. Examples
Problem 20 section 1.7 Determine by inspection whether the
vectors are linearly independent. Justify your answer.
1 −2 0
4 , 5 , 0
−7 3 0
Solution The zero vector is a part of this set. So the set is linearly
dependent.
18. What happened here?
The matrix A acted on a vector x from R4 and produced a
new vector b in R2 . Or, A transforms x into b.
19. What happened here?
The matrix A acted on a vector x from R4 and produced a
new vector b in R2 . Or, A transforms x into b.
The matrix A acted on a vector u from R4 and produced the
zero vector 0 in R2 . Or, A transforms u into 0.
20. What happened here?
The matrix A acted on a vector x from R4 and produced a
new vector b in R2 . Or, A transforms x into b.
The matrix A acted on a vector u from R4 and produced the
zero vector 0 in R2 . Or, A transforms u into 0.
Solving Ax = b is equivalent to nding all vectors x in R4 that are
transformed into the vector b in R2 when acted upon by A
21. Pictorially
Multiplication
by A
b
x
0
0
u Multiplication R2
4
R by A
22. Pictorially
Multiplication
by A
b
x
0
0
u Multiplication R2
4
R by A
23. Pictorially
Multiplication
by A
b
x
0
0
u Multiplication R2
4
R by A
24. Remember functions from calculus?
The correspondence between x and Ax is a function from one set
of vectors to the other.
25. Remember functions from calculus?
The correspondence between x and Ax is a function from one set
of vectors to the other.
Again, function just transforms one real number into another.
26. Transformation, Domain etc
A transformation (or function or mapping) T from Rn to Rm is a
rule that assigns to each vector x in Rn a vector T (x) in Rm .
27. Transformation, Domain etc
A transformation (or function or mapping) T from Rn to Rm is a
rule that assigns to each vector x in Rn a vector T (x) in Rm .
The set Rn is called Domain of T .
28. Transformation, Domain etc
A transformation (or function or mapping) T from Rn to Rm is a
rule that assigns to each vector x in Rn a vector T (x) in Rm .
The set Rn is called Domain of T .
The set Rm is called Co-Domain of T .
29. Transformation, Domain etc
A transformation (or function or mapping) T from Rn to Rm is a
rule that assigns to each vector x in Rn a vector T (x) in Rm .
The set Rn is called Domain of T .
The set Rm is called Co-Domain of T .
The notation T : Rn → Rm means the domain is Rn and the
co-domain is Rm .
30. Transformation, Domain etc
A transformation (or function or mapping) T from Rn to Rm is a
rule that assigns to each vector x in Rn a vector T (x) in Rm .
The set Rn is called Domain of T .
The set Rm is called Co-Domain of T .
The notation T : Rn → Rm means the domain is Rn and the
co-domain is Rm .
For x in Rn , the vector T (x) is called the image of x.
31. Transformation, Domain etc
A transformation (or function or mapping) T from Rn to Rm is a
rule that assigns to each vector x in Rn a vector T (x) in Rm .
The set Rn is called Domain of T .
The set Rm is called Co-Domain of T .
The notation T : Rn → Rm means the domain is Rn and the
co-domain is Rm .
For x in Rn , the vector T (x) is called the image of x.
Set of all images T (x) is called the Range of T .
33. Example 2, Section 1.8
0. 5 0 0
1
a
Let A = 0 0.5 0 , u= 0 , v = b
0 0 0.5 −4 c
Dene T : R3 → R3 by T (x) = Ax. Find T (u) and T (v).
34. Example 2, Section 1.8
0. 5 0 0
1
a
Let A = 0 0.5 0 , u= 0 , v = b
0 0 0.5 −4 c
Dene T : R3 → R3 by T (x) = Ax. Find T (u) and T (v).
Solution The problem is just asking you to nd the products Au
and Av
(0.5)(1) + 0.0 + 0.(−4) 0. 5
Au = 0.1 + (0.5).0 + 0.(−4) = 0
0.1 + 0.0 + (0.5).(−4) −2
(0.5)(a) + 0.b + 0.c 0. 5a
Av = 0.a + (0.5).b + 0.c = 0.5b
0.a + 0.b + (0.5).c 0. 5c
35. Example 4, Section 1.8
1 −3 2 6
Let A = 0 1 4 , b= −7
3 −5 −9 −9
Let T be dened by by T (x) = Ax. Find a vector x whose image
under T is b and determine whether x is unique.
36. Example 4, Section 1.8
1 −3 2 6
Let A = 0 1 4 , b= −7
3 −5 −9 −9
Let T be dened by by T (x) = Ax. Find a vector x whose image
under T is b and determine whether x is unique.
Solution The problem is asking you to solve Ax = b. In other words,
write the augmented matrix and solve.
1 −3 2 6
R3-3R1
0 1 −4 −7
3 −5 −9 −9
39. Example 4, Section 1.8
1 0 0 −5
0 1 0 −3
0 0 1 1
−5
So, x = −3.
1
40. Example 4, Section 1.8
1 0 0 −5
0 1 0 −3
0 0 1 1
−5
So, x = −3. All columns have pivots which means no free
1
variables. So the solution is unique.