The document discusses the composition of linear transformations. It defines the composition of two linear transformations T1 and T2 as the transformation T2 o T1, which maps an element X in the domain of T1 to the result of applying T2 to the output of T1. The key points made are:
1) The composition T2 o T1 is a linear transformation.
2) The matrix associated with the composition T2 o T1 is the product of the matrices associated with T1 and T2.
3) Examples are provided to illustrate finding the matrix of a composition and applying a composition to vectors and subspaces.
This document discusses matrix representations of linear transformations and changes of basis in linear algebra. It defines the matrix associated with a linear transformation with respect to two bases and introduces the change of basis matrices. It provides examples of finding the matrix associated with a linear transformation, the change of basis matrices between two bases, and using the change of basis matrices to transform component representations of a vector between bases.
This document discusses theorems related to linear transformations between finite-dimensional vector spaces. It proves two main theorems:
1) A linear transformation T is invertible if and only if T maps a basis of the domain space V to a basis of the codomain space W.
2) A linear transformation T between vector spaces of equal dimension is invertible, injective, surjective, and maps bases to bases. These properties are shown to be equivalent.
The document provides a detailed proof of each theorem with examples to illustrate the concepts. It discusses key ideas such as linear independence, spanning sets, and the relationship between invertibility, injectivity and surjectivity of linear transformations.
This document discusses the matrix associated with a linear transformation. It defines key concepts such as:
1) The matrix associated with a linear transformation T between vector spaces V and W depends on the chosen bases [v] of V and [w] of W.
2) The entries of the matrix are the coefficients that relate the images of the basis vectors of [v] under T in terms of the basis vectors of [w].
3) The matrix allows representing the transformation T of any vector in V in terms of a matrix multiplication between the vector's coordinates in [v] and the transformation matrix.
This document discusses two theorems: Gauss divergence theorem and Stokes theorem. It provides an example problem to verify each theorem. For Gauss divergence theorem, it calculates the surface integral of a vector function over the surfaces of a rectangle parallelepiped and shows it equals the volume integral of the divergence of the function over the volume, verifying the theorem. For Stokes theorem, it similarly calculates line and surface integrals of a vector function over a rectangular region to verify the theorem holds.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIRai University
This document provides an overview of Unit II - Complex Integration from the Engineering Mathematics-IV course at RAI University, Ahmedabad. It covers key topics such as:
1) Complex line integrals and Cauchy's integral theorem which states that the integral of an analytic function around a closed curve is zero.
2) Cauchy's integral formula which can be used to evaluate integrals and find derivatives of analytic functions.
3) Taylor and Laurent series expansions of functions, including their regions of convergence.
4) The residue theorem which can be used to evaluate real integrals involving trigonometric or rational functions.
B.tech ii unit-4 material vector differentiationRai University
This document discusses concepts in vector differentiation including:
1. It defines scalar and vector point functions, with examples such as temperature and fluid velocity.
2. It introduces the vector differential operator del (∇) and defines the gradient of a scalar function.
3. It explains normal and directional derivatives, and gives examples of finding the gradient and directional derivative of functions.
4. It defines the divergence of a vector function and gives examples of applying vector differentiation concepts.
This document provides an overview of vector differentiation, including gradient, divergence, curl, and related concepts. It begins with definitions of scalar and vector point functions. It then defines the vector differential operator Del and explores using it to calculate the gradient of a scalar function, directional derivatives, and normal derivatives. The document also covers divergence and curl, providing their definitions and formulas. Examples are given for calculating gradient, divergence, curl, and directional derivatives. The document concludes with exercises and references for further reading.
This document discusses vector differentiation and provides examples and exercises. Some key topics covered include:
- The definition of scalar and vector point functions
- Vector differential operator Del (∇)
- Gradient of a scalar function and its properties
- Normal and directional derivatives
- Divergence and curl of vector functions
- Examples calculating gradients, directional derivatives, and verifying identities
- Exercises involving finding normals, gradients, directional derivatives, and divergence
This document discusses matrix representations of linear transformations and changes of basis in linear algebra. It defines the matrix associated with a linear transformation with respect to two bases and introduces the change of basis matrices. It provides examples of finding the matrix associated with a linear transformation, the change of basis matrices between two bases, and using the change of basis matrices to transform component representations of a vector between bases.
This document discusses theorems related to linear transformations between finite-dimensional vector spaces. It proves two main theorems:
1) A linear transformation T is invertible if and only if T maps a basis of the domain space V to a basis of the codomain space W.
2) A linear transformation T between vector spaces of equal dimension is invertible, injective, surjective, and maps bases to bases. These properties are shown to be equivalent.
The document provides a detailed proof of each theorem with examples to illustrate the concepts. It discusses key ideas such as linear independence, spanning sets, and the relationship between invertibility, injectivity and surjectivity of linear transformations.
This document discusses the matrix associated with a linear transformation. It defines key concepts such as:
1) The matrix associated with a linear transformation T between vector spaces V and W depends on the chosen bases [v] of V and [w] of W.
2) The entries of the matrix are the coefficients that relate the images of the basis vectors of [v] under T in terms of the basis vectors of [w].
3) The matrix allows representing the transformation T of any vector in V in terms of a matrix multiplication between the vector's coordinates in [v] and the transformation matrix.
This document discusses two theorems: Gauss divergence theorem and Stokes theorem. It provides an example problem to verify each theorem. For Gauss divergence theorem, it calculates the surface integral of a vector function over the surfaces of a rectangle parallelepiped and shows it equals the volume integral of the divergence of the function over the volume, verifying the theorem. For Stokes theorem, it similarly calculates line and surface integrals of a vector function over a rectangular region to verify the theorem holds.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIRai University
This document provides an overview of Unit II - Complex Integration from the Engineering Mathematics-IV course at RAI University, Ahmedabad. It covers key topics such as:
1) Complex line integrals and Cauchy's integral theorem which states that the integral of an analytic function around a closed curve is zero.
2) Cauchy's integral formula which can be used to evaluate integrals and find derivatives of analytic functions.
3) Taylor and Laurent series expansions of functions, including their regions of convergence.
4) The residue theorem which can be used to evaluate real integrals involving trigonometric or rational functions.
B.tech ii unit-4 material vector differentiationRai University
This document discusses concepts in vector differentiation including:
1. It defines scalar and vector point functions, with examples such as temperature and fluid velocity.
2. It introduces the vector differential operator del (∇) and defines the gradient of a scalar function.
3. It explains normal and directional derivatives, and gives examples of finding the gradient and directional derivative of functions.
4. It defines the divergence of a vector function and gives examples of applying vector differentiation concepts.
This document provides an overview of vector differentiation, including gradient, divergence, curl, and related concepts. It begins with definitions of scalar and vector point functions. It then defines the vector differential operator Del and explores using it to calculate the gradient of a scalar function, directional derivatives, and normal derivatives. The document also covers divergence and curl, providing their definitions and formulas. Examples are given for calculating gradient, divergence, curl, and directional derivatives. The document concludes with exercises and references for further reading.
This document discusses vector differentiation and provides examples and exercises. Some key topics covered include:
- The definition of scalar and vector point functions
- Vector differential operator Del (∇)
- Gradient of a scalar function and its properties
- Normal and directional derivatives
- Divergence and curl of vector functions
- Examples calculating gradients, directional derivatives, and verifying identities
- Exercises involving finding normals, gradients, directional derivatives, and divergence
B.tech ii unit-3 material multiple integrationRai University
1. The document discusses multiple integrals and double integrals. It defines double integrals and provides two methods for evaluating them: integrating first with respect to one variable and then the other, or vice versa.
2. Examples are given of evaluating double integrals using these methods over different regions of integration in the xy-plane, including integrals over a circle and a hyperbolic region.
3. The document also discusses calculating double integrals over a region when the limits of integration are not explicitly given, but the region is described geometrically.
This document discusses Hermite polynomials and their properties. It begins by introducing the Hermite equation, which arises from the theory of the linear harmonic oscillator. The Hermite equation is then solved using a series solution approach. Explicit formulas for the Hermite polynomials Hn(x) are derived for n=0,1,2,3,... using properties of the Hermite equation like recursion relations. Key properties of the Hermite polynomials like generating functions, even/odd behavior, Rodrigue's formula, orthogonality, and recurrence relations are also proved.
B.tech ii unit-5 material vector integrationRai University
This document discusses various vector integration topics:
1. It defines line, surface, and volume integrals and provides examples of evaluating each. Line integrals deal with vector fields along paths, surface integrals deal with vector fields over surfaces, and volume integrals deal with vector fields throughout a volume.
2. Green's theorem, Stokes' theorem, and Gauss's theorem are introduced as relationships between these types of integrals but their proofs are not shown.
3. Examples are provided to demonstrate evaluating line integrals of conservative and non-conservative vector fields, as well as a surface integral over a spherical surface.
This document discusses vectors and tensors in three dimensions. It defines reciprocal sets of vectors, which satisfy certain orthogonality properties. It introduces the metric tensor or fundamental tensor, which is specified by three non-coplanar vectors. It proves that the determinant of the metric tensor, known as the Gram determinant, is non-zero using properties of the defining vectors.
The document defines the gamma and beta functions and provides examples of using them to evaluate integrals. The gamma function Γ(n) generalizes the factorial function to real and complex numbers. It satisfies properties like Γ(n+1)=nΓ(n). The beta function B(m,n) defines integrals over the interval [0,1]. It relates to the gamma function as B(m,n)=Γ(m)Γ(n)/Γ(m+n). Several integrals are evaluated using these functions, including changing variables to match their definitions. Proofs are also given for relationships between beta function integrals over [0,1] and [0,π/2].
This document provides an overview of vector integration, including line integrals, surface integrals, and volume integrals. It defines each type of integral and provides examples of evaluating them. For line integrals, it discusses work, circulation, and path independence. Surface integrals are defined as the integral of the normal component of a vector field over an enclosed surface. Volume integrals integrate a vector field over a three-dimensional volume. Worked examples demonstrate evaluating specific line, surface, and volume integrals.
This document discusses the gamma and beta functions. It defines the gamma function and lists some of its key properties. Examples are provided to demonstrate how to evaluate integrals using gamma function properties. The beta function is then defined and its relationship to the gamma function explained. Dirichlet's integral theorem and its extension to multiple dimensions is covered. Applications to finding volumes and masses are demonstrated. References for further reading on gamma and beta functions are listed at the end.
This unit covers the formation and solutions of partial differential equations (PDEs). PDEs can be obtained by eliminating arbitrary constants or functions from relating equations. Standard methods are used to solve first order PDEs and higher order linear PDEs with constant coefficients. Various physical processes are modeled using PDEs including the wave equation, heat equation, and Laplace's equation.
This document provides lecture notes on complex analysis covering four units of content:
1) The index of a close curve, Cauchy's theorem, and entire functions.
2) Counting zeroes, meromorphic functions, and maximum principle.
3) Spaces of continuous and analytic functions, and behavior of functions.
4) Comparison of entire functions, analytic continuation, and harmonic functions.
It also provides definitions and theorems regarding integrals over rectifiable curves, winding numbers, and Cauchy's theorem. Exercises and proofs are included.
1) Advanced metal forming techniques involve plastic deformation of metal crystals beyond their elastic limit.
2) Yield criteria determine which combination of multi-axial stresses will cause yielding, such as Tresca and von-Mises criteria relating to maximum shear stresses.
3) Mohr's circle provides a graphical representation of the state of stress at a point, showing normal and shear stresses on planes of all orientations. It can be used to determine principal stresses and maximum shear stress.
This document discusses numerical differentiation and integration using Newton's forward and backward difference formulas. It provides examples of using these formulas to calculate derivatives from tables of ordered data pairs. Specifically, it shows how to calculate derivatives at interior points using central difference formulas, and at endpoints using forward or backward formulas depending on if the point is near the start or end of the data range. Formulas are derived for calculating the first and second derivatives, and examples are worked through to find acceleration and rates of cooling from given temperature-time tables.
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.
This document provides an overview of techniques for solving advanced engineering mathematics problems involving higher order differential equations. It introduces key concepts like auxiliary equations, complementary functions, particular integrals, and methods for solving linear differential equations with constant and variable coefficients. These include the general method, shortcut method, method of undetermined coefficients, and method of variation parameters (Wronskian method). Examples of applying these techniques to solve specific differential equations are also provided.
The document discusses finite difference methods for solving differential equations. It begins by introducing finite difference methods as alternatives to shooting methods for solving differential equations numerically. It then provides details on using finite difference methods to transform differential equations into algebraic equations that can be solved. This includes deriving finite difference approximations for derivatives, setting up the finite difference equations at interior points, and assembling the equations in matrix form. The document also provides an example of applying a finite difference method to solve a linear boundary value problem and a nonlinear boundary value problem.
The document provides an overview of advanced engineering mathematics concepts for differential equations. It covers topics such as homogeneous and non-homogeneous linear differential equations with constant and variable coefficients. Methods for solving differential equations are discussed, including finding the auxiliary equation, complementary function, and particular integral. Specific solving techniques like the Cauchy-Euler and Legendre methods for variable coefficient equations are also mentioned. Examples of different types of differential equations are provided throughout.
This document contains solutions to 4 problems regarding Cauchy sequences:
1) It provides an example of a bounded sequence that is not Cauchy by considering the sequence {(-1)^n}.
2) It shows that the sequences (n+1/n) and (1 + 1/2! + ... + 1/n!) are Cauchy using the definition.
3) It shows that the sequences ((-1)^n), (n + (-1)^n/n), and (ln(n)) are not Cauchy by finding values that violate the definition.
4) It proves that if (x_n) and (y_n) are Cauchy, then (x_n +
This document provides an overview of topics in vector integration, including line integrals, surface integrals, and volume integrals. It includes examples of calculating each type of integral. The key theorems covered are Green's theorem, Stokes' theorem, and Gauss's theorem of divergence. Green's theorem relates a line integral around a closed curve to a double integral over the enclosed region. Stokes' theorem relates a line integral around a closed curve to a surface integral over the enclosed surface. Gauss's theorem relates the surface integral of the normal component of a vector field over a closed surface to the volume integral of the divergence of the vector field over the enclosed volume.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IRai University
1. The document discusses functions of complex variables, including analytic functions, Cauchy-Riemann equations, harmonic functions, and methods for determining an analytic function when its real or imaginary part is known.
2. Some key topics covered are the definition of an analytic function, Cauchy-Riemann equations in Cartesian and polar forms, properties of analytic functions including orthogonal systems, and determining the analytic function using methods like direct, Milne-Thomson's, and exact differential equations.
3. Examples are provided to illustrate determining the analytic function given its real or imaginary part, such as finding the function when the real part is a polynomial or the imaginary part is a trigonometric function.
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
B.tech ii unit-3 material multiple integrationRai University
1. The document discusses multiple integrals and double integrals. It defines double integrals and provides two methods for evaluating them: integrating first with respect to one variable and then the other, or vice versa.
2. Examples are given of evaluating double integrals using these methods over different regions of integration in the xy-plane, including integrals over a circle and a hyperbolic region.
3. The document also discusses calculating double integrals over a region when the limits of integration are not explicitly given, but the region is described geometrically.
This document discusses Hermite polynomials and their properties. It begins by introducing the Hermite equation, which arises from the theory of the linear harmonic oscillator. The Hermite equation is then solved using a series solution approach. Explicit formulas for the Hermite polynomials Hn(x) are derived for n=0,1,2,3,... using properties of the Hermite equation like recursion relations. Key properties of the Hermite polynomials like generating functions, even/odd behavior, Rodrigue's formula, orthogonality, and recurrence relations are also proved.
B.tech ii unit-5 material vector integrationRai University
This document discusses various vector integration topics:
1. It defines line, surface, and volume integrals and provides examples of evaluating each. Line integrals deal with vector fields along paths, surface integrals deal with vector fields over surfaces, and volume integrals deal with vector fields throughout a volume.
2. Green's theorem, Stokes' theorem, and Gauss's theorem are introduced as relationships between these types of integrals but their proofs are not shown.
3. Examples are provided to demonstrate evaluating line integrals of conservative and non-conservative vector fields, as well as a surface integral over a spherical surface.
This document discusses vectors and tensors in three dimensions. It defines reciprocal sets of vectors, which satisfy certain orthogonality properties. It introduces the metric tensor or fundamental tensor, which is specified by three non-coplanar vectors. It proves that the determinant of the metric tensor, known as the Gram determinant, is non-zero using properties of the defining vectors.
The document defines the gamma and beta functions and provides examples of using them to evaluate integrals. The gamma function Γ(n) generalizes the factorial function to real and complex numbers. It satisfies properties like Γ(n+1)=nΓ(n). The beta function B(m,n) defines integrals over the interval [0,1]. It relates to the gamma function as B(m,n)=Γ(m)Γ(n)/Γ(m+n). Several integrals are evaluated using these functions, including changing variables to match their definitions. Proofs are also given for relationships between beta function integrals over [0,1] and [0,π/2].
This document provides an overview of vector integration, including line integrals, surface integrals, and volume integrals. It defines each type of integral and provides examples of evaluating them. For line integrals, it discusses work, circulation, and path independence. Surface integrals are defined as the integral of the normal component of a vector field over an enclosed surface. Volume integrals integrate a vector field over a three-dimensional volume. Worked examples demonstrate evaluating specific line, surface, and volume integrals.
This document discusses the gamma and beta functions. It defines the gamma function and lists some of its key properties. Examples are provided to demonstrate how to evaluate integrals using gamma function properties. The beta function is then defined and its relationship to the gamma function explained. Dirichlet's integral theorem and its extension to multiple dimensions is covered. Applications to finding volumes and masses are demonstrated. References for further reading on gamma and beta functions are listed at the end.
This unit covers the formation and solutions of partial differential equations (PDEs). PDEs can be obtained by eliminating arbitrary constants or functions from relating equations. Standard methods are used to solve first order PDEs and higher order linear PDEs with constant coefficients. Various physical processes are modeled using PDEs including the wave equation, heat equation, and Laplace's equation.
This document provides lecture notes on complex analysis covering four units of content:
1) The index of a close curve, Cauchy's theorem, and entire functions.
2) Counting zeroes, meromorphic functions, and maximum principle.
3) Spaces of continuous and analytic functions, and behavior of functions.
4) Comparison of entire functions, analytic continuation, and harmonic functions.
It also provides definitions and theorems regarding integrals over rectifiable curves, winding numbers, and Cauchy's theorem. Exercises and proofs are included.
1) Advanced metal forming techniques involve plastic deformation of metal crystals beyond their elastic limit.
2) Yield criteria determine which combination of multi-axial stresses will cause yielding, such as Tresca and von-Mises criteria relating to maximum shear stresses.
3) Mohr's circle provides a graphical representation of the state of stress at a point, showing normal and shear stresses on planes of all orientations. It can be used to determine principal stresses and maximum shear stress.
This document discusses numerical differentiation and integration using Newton's forward and backward difference formulas. It provides examples of using these formulas to calculate derivatives from tables of ordered data pairs. Specifically, it shows how to calculate derivatives at interior points using central difference formulas, and at endpoints using forward or backward formulas depending on if the point is near the start or end of the data range. Formulas are derived for calculating the first and second derivatives, and examples are worked through to find acceleration and rates of cooling from given temperature-time tables.
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.
This document provides an overview of techniques for solving advanced engineering mathematics problems involving higher order differential equations. It introduces key concepts like auxiliary equations, complementary functions, particular integrals, and methods for solving linear differential equations with constant and variable coefficients. These include the general method, shortcut method, method of undetermined coefficients, and method of variation parameters (Wronskian method). Examples of applying these techniques to solve specific differential equations are also provided.
The document discusses finite difference methods for solving differential equations. It begins by introducing finite difference methods as alternatives to shooting methods for solving differential equations numerically. It then provides details on using finite difference methods to transform differential equations into algebraic equations that can be solved. This includes deriving finite difference approximations for derivatives, setting up the finite difference equations at interior points, and assembling the equations in matrix form. The document also provides an example of applying a finite difference method to solve a linear boundary value problem and a nonlinear boundary value problem.
The document provides an overview of advanced engineering mathematics concepts for differential equations. It covers topics such as homogeneous and non-homogeneous linear differential equations with constant and variable coefficients. Methods for solving differential equations are discussed, including finding the auxiliary equation, complementary function, and particular integral. Specific solving techniques like the Cauchy-Euler and Legendre methods for variable coefficient equations are also mentioned. Examples of different types of differential equations are provided throughout.
This document contains solutions to 4 problems regarding Cauchy sequences:
1) It provides an example of a bounded sequence that is not Cauchy by considering the sequence {(-1)^n}.
2) It shows that the sequences (n+1/n) and (1 + 1/2! + ... + 1/n!) are Cauchy using the definition.
3) It shows that the sequences ((-1)^n), (n + (-1)^n/n), and (ln(n)) are not Cauchy by finding values that violate the definition.
4) It proves that if (x_n) and (y_n) are Cauchy, then (x_n +
This document provides an overview of topics in vector integration, including line integrals, surface integrals, and volume integrals. It includes examples of calculating each type of integral. The key theorems covered are Green's theorem, Stokes' theorem, and Gauss's theorem of divergence. Green's theorem relates a line integral around a closed curve to a double integral over the enclosed region. Stokes' theorem relates a line integral around a closed curve to a surface integral over the enclosed surface. Gauss's theorem relates the surface integral of the normal component of a vector field over a closed surface to the volume integral of the divergence of the vector field over the enclosed volume.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IRai University
1. The document discusses functions of complex variables, including analytic functions, Cauchy-Riemann equations, harmonic functions, and methods for determining an analytic function when its real or imaginary part is known.
2. Some key topics covered are the definition of an analytic function, Cauchy-Riemann equations in Cartesian and polar forms, properties of analytic functions including orthogonal systems, and determining the analytic function using methods like direct, Milne-Thomson's, and exact differential equations.
3. Examples are provided to illustrate determining the analytic function given its real or imaginary part, such as finding the function when the real part is a polynomial or the imaginary part is a trigonometric function.
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
We disclose a simple and straightforward method of solving ordinary or linear partial differential equations of any order and apply it to solve the generalized Euler-Tricomi equation. The method is easier than classical methods and also didactic.
Date: Jan, 10, 202
This document is an internship project report submitted by Siddharth Pujari to the Indian Institute of Space Science and Technology. The report focuses on advanced control system design for aircraft and simulating aircraft trajectory. It includes modeling an aircraft's state space model in MATLAB to test controllability. The report also covers theoretical aspects of stability of linear systems, linearizing nonlinear models, controllability of linear systems using the Kalman criterion and transition matrix, and applying these concepts to simulate aircraft controllability in MATLAB.
We present a strong convergence implicit Runge-Kutta method, with four stages, for solution of
initial value problem of ordinary differential equations. Collocation method is used to derive a continuous
scheme; and the continuous scheme evaluated at special points, the Gaussian points of fourth degree Legendre
polynomial, gives us four function evaluations and the Runge-Kutta method for the iteration of the solutions.
Convergent properties of the method are discussed. Experimental problems used to check the quality of the
scheme show that the method is highly efficient, A – stable, has simple structure, converges to exact solution
faster and better than some existing popular methods cited in this paper.
The document discusses applications of integration, including calculating the length of a curve and surface area of solids obtained by rotating curves. It provides formulas for finding the arc length of a curve given by y=f(x), and surface area of solids obtained by rotating curves about the x- or y-axes. Examples are worked out applying these formulas to find the arc length of curves and surface area of rotated regions. The document also discusses evaluating triple integrals to find the volume of a three-dimensional region and using triple integrals to find the centroid of a volume.
Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_n│a_n=n^3,n∈N}, B3={b_n│b_n=a_(n+1)-a_n }, C3={c_n│c_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3∧x^3∈A3, y∈B3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascal’s rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
This document provides an overview of analytic functions in engineering mathematics. It defines analytic functions as functions whose derivatives exist in some neighborhood of a point, making them continuously differentiable. The Cauchy-Riemann equations are derived as necessary conditions for a function to be analytic. It also defines entire functions as analytic functions over the entire finite plane. Examples of entire functions include exponential, sine, cosine, and hyperbolic functions. The document discusses analyticity in both Cartesian and polar coordinates.
The document discusses systems of two variable equations and inequalities. It defines a system of two variable equations as a collection of two or more equations involving two variables (linear-linear, linear-quadratic, quadratic-quadratic) whose solutions are points satisfying the equations. The solution graph is the intersection of the equations. It also defines systems of two variable inequalities and explains how to graphically represent the solution sets. Examples of solving systems using graphical and substitution methods are provided.
1) The document discusses parallel summable range symmetric matrices over an incline. It defines what it means for two matrices to be parallel summable and defines their parallel sum.
2) It proves several properties of parallel summable matrices, including that the sum and parallel sum of parallel summable range symmetric matrices are also range symmetric.
3) It establishes conditions under which the sum or parallel sum of two matrices will be range symmetric, such as if their row and column ranges are properly contained within each other.
This document discusses the Laplace transform, which is a mathematical technique that can be used to solve linear differential equations. It defines the Laplace transform of a function f(t) as the integral from 0 to infinity of e^-st f(t) dt, where s is a parameter. The document lists sufficient conditions for a function's Laplace transform to exist and provides several properties of the Laplace transform, including linearity, scaling properties, shifting properties, and transforms of derivatives and integrals. It gives examples to illustrate these concepts.
Ejercicios resueltos de analisis matematico 1tinardo
The document describes the logarithmic differentiation method used to derive functions where the exponent is a variable. It explains the steps: take the natural log of both sides, apply logarithm properties, derive both terms, isolate the function, and substitute back in. Examples are provided and solved, such as deriving y=xx, y=sen(x)(x3+6x), and y=ln x3 + 5x2cos(x). Related activities are summarized with solutions to practice problems applying this method.
Matrix Transformations on Some Difference Sequence SpacesIOSR Journals
The sequence spaces 𝑙∞(𝑢,𝑣,Δ), 𝑐0(𝑢,𝑣,Δ) and 𝑐(𝑢,𝑣,Δ) were recently introduced. The matrix classes (𝑐 𝑢,𝑣,Δ :𝑐) and (𝑐 𝑢,𝑣,Δ :𝑙∞) were characterized. The object of this paper is to further determine the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (𝑐 𝑢,𝑣,Δ ∶𝑏𝑠) and (𝑐 𝑢,𝑣,Δ ∶ 𝑙𝑝). It is observed that the later characterizations are additions to the existing ones
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Very brief highlights on some key details 2foxtrot jp R
This document provides a summary of key details regarding vacuum-to-vacuum matrix calculations in the presence of cubic self-interaction terms. It begins by defining the initial and evolved vacuum states, and the vacuum-to-vacuum matrix for the probability amplitude of remaining in the vacuum state. It then introduces the time evolution operator and scalar field Hamiltonian. The document derives expressions for the vacuum-to-vacuum matrix including cubic self-interactions, translating the expressions into momentum space. It provides Feynman diagrams corresponding to the momentum space expressions.
This document provides a summary of key linear algebra concepts including:
1. Representing systems of linear equations using matrices and solving them through row reduction operations.
2. Calculating matrix products, inverses, determinants, eigenvalues, and eigenvectors.
3. Performing singular value decompositions (SVD) which generalizes eigendecomposition to non-square matrices by finding orthogonal matrices and singular values that factorize a matrix.
Fixed Point Results for Weakly Compatible Mappings in Convex G-Metric Spaceinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
graphs of quadratic function grade 9.pptxMeryAnnMAlday
This document discusses quadratic functions and their graphs. It defines key features of quadratic graphs including the vertex, axis of symmetry, and effect of coefficients on the opening of the parabola. Examples are provided to demonstrate how to identify these features from the equation, generate tables of values, and graph the quadratic function. Learning targets are outlined to draw and analyze quadratic graphs and the effect of changing coefficients on the graph.
Differential Geometry for Machine LearningSEMINARGROOT
References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
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1. CAPÍTULO III
MATRIZ ASOCIADA A UNA TRANSFORMACIÓN LINEAL.
CONTENIDO:
3.1 Matriz asociada a una transformación lineal.
3.2 Algebra de transformaciones lineales.
3.3 Composición de Transformaciones lineales.
3.4 Transformaciones lineales invertibles.
3.5 Teorema de equivalencias de una transformación lineal.
3.6 Isomorfismo inducido por una transformación lineal.
3.7 Cambio de base y semejanza de matrices.
3.8 Producto Interno. Definición. Teoremas de caracterización.
3.9 Norma de un Vector.
3.10 Ortogonalidad. Conjunto ortogonal y conjunto ortonormal.
3.11 Proceso de ortogonalidad de Gram Schmitdt.
3.12 Espacio Dual de un espacio vectorial.
3.13 Adjunta de una transformación lineal
2. 88
3.3 COMPOSICIÓN DE TRANSFORMACIONES LINEALES. O PRODUCTO DE
TRANSFORMACIONES LINEALES.
DEFINICIÓN.Sean(𝑉;𝐾,+; ∙),(𝑊;𝐾,+; ∙) y (𝑈; 𝐾, +; ∙) espacios vectoriales y sean 𝑇1: 𝑉 → 𝑊y
𝑇2:𝑊 → 𝑈 dos transformaciones lineales. La aplicación composición 𝑇2o𝑇1:𝑉 → 𝑈 se define
como (𝑇2o𝑇1)(𝑋) = 𝑇2[𝑇1(𝑋)], ∀ 𝑋 ∈ 𝑉.
NOTA 1:La expresión 𝑇2o𝑇1 se lee: “Composición de 𝑇1 con 𝑇2” o también “𝑇2 compuesta con
𝑇1”
NOTA 2: La composición se puede generalizar para todas las transformaciones lineales entre
los espacios vectoriales: 𝑇𝑖 ∈ 𝐿(𝑉;𝑊) y 𝑇𝑗 ∈ 𝐿(𝑊;𝑈)
OBSERVACIÓN.La composición 𝑇2o𝑇1 es posible cuando la dimensión del espacio de llegada de
𝑇1 es igual a la dimensión del espacio de partida de 𝑇2 .
La composición 𝑇1o𝑇2 es posible cuando la dimensión del espacio de llegada de 𝑇2 es igual a la
dimensión del espacio de partida de 𝑇1 .
(𝑉; 𝐾, +; ∙) (𝑊;𝐾,+; ∙) (𝑈;𝐾,+; ∙)
𝑋 . 𝑇1(𝑋) . . 𝑇2[𝑇1(𝑋)] = (𝑇2o 𝑇1 )(𝑋)
𝑇2o 𝑇1
𝑇2
𝑇1
>
>
(𝑉; 𝐾, +; ∙) (𝑊;𝐾,+; ∙) (𝑈;𝐾,+; ∙)
𝑋 . 𝑇𝑖(𝑋) . . 𝑇𝑗[𝑇𝑖(𝑋)] = 𝑇𝑗o 𝑇𝑖 (𝑋)
𝑇𝑗o 𝑇𝑖
𝑇𝑗
𝑇𝑖
>
>
(𝑉𝑛
; 𝐾+; ∙) (𝑊𝑚
; 𝐾+; ∙) (𝑈𝑝
;𝐾+; ∙)
𝑇2𝑜𝑇1
𝑇1 𝑇2
(𝑉𝑛
; 𝐾+; ∙) (𝑊𝑚
; 𝐾+; ∙) (𝑈𝑝
;𝐾+; ∙)
𝑇𝑗𝑜𝑇𝑖
𝑇𝑖 𝑇𝑗
3. 89
TEOREMA. La aplicación composición de dos transformaciones lineales 𝑇2o 𝑇1 es una
transformación lineal.
Demostración.
Sean las transformaciones lineales dadas en la definición y sea 𝑘1; 𝑘2 ∈ 𝐾 y 𝑋1; 𝑋2 ∈ 𝑉.
(𝑇2o 𝑇1)(𝑘1𝑋1 + 𝑘2𝑋2) = 𝑘1(𝑇2o 𝑇1)(𝑋1 ) + 𝑘2(𝑇2o 𝑇1)(𝑋2) ¡Probar!
En efecto:
(𝑇2o 𝑇1)(𝑘1𝑋1 + 𝑘2𝑋2)
= 𝑇2[𝑇1(𝑘1𝑋1 + 𝑘2𝑋2)] Por definición de composición
= 𝑇2[𝑇1(𝑘1𝑋1) + 𝑇1(𝑘2𝑋2)] Pues 𝑇1 es una T. L.
= 𝑇2[𝑘1𝑇1(𝑋1) + 𝑘2𝑇1(𝑋2)] Pues 𝑇1 es una T. L.
= 𝑇2[𝑘1𝑇1(𝑋1)] + 𝑇2[𝑘2𝑇1(𝑋2)] Pues 𝑇2 es una T. L.
= 𝑘1𝑇2[𝑇1(𝑋1)] + 𝑘2𝑇2[𝑇1(𝑋2)] Pues 𝑇2 es una T. L.
= 𝑘1(𝑇2o 𝑇1)(𝑋1) + 𝑘2(𝑇2o 𝑇1)(𝑋2) Por definición de Composición
Por lo tanto, la composición 𝑇2o 𝑇1 es una transformación lineal.
PROPOSICIÓN.Lamatrizasociadaalacomposicióndelastransformacioneslineales𝑇2o𝑇1:𝑉 →
𝑈, siendo 𝑇1:𝑉 → 𝑊 con matriz asociada 𝐴𝑇
1
y 𝑇2:𝑊 → 𝑈 con matriz asociada es dada 𝐴𝑇2
; es
dada por la matriz producto 𝐴𝑇2𝑜𝑇1
= [𝐴𝑇
2
][𝐴𝑇1
]
Prueba. Ejercicio.
PROPOSICIÓN. Para toda 𝑇𝑖 ∈ 𝐿(𝑉; 𝑊) y para todas las 𝑇𝑗 ;𝑇𝑚 ∈ 𝐿(𝑊;𝑍) se cumple que:
𝑇𝑗 + 𝑇𝑚 𝑜𝑇𝑖 = 𝑇𝑗𝑜𝑇𝑖 + 𝑇𝑚𝑜𝑇𝑖
(𝑉𝑛
; 𝐾+; ∙) (𝑊𝑚
; 𝐾+; ∙) (𝑈𝑝
;𝐾+; ∙)
𝑇1𝑜𝑇2
𝑇2 𝑇1
(𝑉𝑛
; 𝐾+; ∙) (𝑊𝑚
; 𝐾+; ∙) (𝑈𝑝
;𝐾+; ∙)
𝑇𝑖𝑜𝑇𝑗
𝑇𝑗 𝑇𝑖
4. 90
Prueba. Ejercicio.
PROPOSICIÓN. Para todas las 𝑇𝑖 ; 𝑇𝑗 ∈ 𝐿(𝑉;𝑊) y para toda 𝑇𝑚 ∈ 𝐿(𝑊;𝑍) se cumple que:
𝑇𝑚𝑜 𝑇𝑖 + 𝑇𝑗 = 𝑇𝑚𝑜𝑇𝑖 + 𝑇𝑚𝑜𝑇𝑗
Prueba. Ejercicio.
PROPOSICIÓN.Paratoda𝑟 ∈ 𝐾;paratoda𝑇𝑖 ∈ 𝐿(𝑉;𝑊)y paratoda𝑇𝑗 ∈ 𝐿(𝑊; 𝑍)secumpleque:
a) 𝑟(𝑇𝑖𝑜𝑇𝑗) = (𝑟𝑇𝑖)𝑜𝑇𝑗
a) 𝑟(𝑇𝑖𝑜𝑇𝑗) = 𝑇𝑖𝑜 𝑟𝑇𝑗
Prueba. Ejercicio.
PROPOSICIÓN. Para toda 𝑇𝑖 ∈ 𝐿(𝑉; 𝑊); para toda 𝑇𝑗 ∈ 𝐿(𝑊;𝑍) y para toda 𝑇𝑘 ∈ 𝐿(𝑍;𝑈) se
cumple que: 𝑇𝑘𝑜 𝑇𝑗𝑜𝑇𝑖 = 𝑇𝑘𝑜𝑇𝑗 𝑜𝑇𝑖
Prueba. Ejercicio.
EJEMPLO 1. Sea 𝑇1:𝑅2 → 𝑅3 tal que 𝑇1(𝑥;𝑦) = (𝑥;𝑥 − 𝑦; 𝑦) y sea 𝑇2:𝑅3 → 𝑅2 tal que
𝑇2(𝑥; 𝑦;𝑧) = (𝑥 + 𝑧;𝑦).
a) Hallar si es posible 𝑇2o 𝑇1
b) Aplique 𝑇2o 𝑇1 al vector 𝑢 = (3;1)
c) Aplique 𝑇2o 𝑇1 al subespacio vectorial 𝑆 = {(𝑎; 𝑏) ∈ 𝑅2 𝑏 = 2𝑎
⁄ }
d) Hallar sus matrices asociadas a 𝑇1 , 𝑇2 y 𝑇2𝑜𝑇1
Solución.
a) La transformación lineal composición 𝑇2o 𝑇1 si es posible como se ve en seguida:
Siendo: 𝑇1(𝑥; 𝑦) = (𝑥;𝑥 − 𝑦; 𝑦) ; 𝑇2(𝑥;𝑦;𝑧) = (𝑥 + 𝑧;𝑦)
(𝑅2;𝑅; +; ∙) (𝑅3;𝑅; +; ∙) (𝑅2;𝑅; +; ∙)
𝑢 . 𝑇1(𝑢) . . 𝑇2[𝑇1(𝑢)] = (𝑇2o 𝑇1)(𝑢)
𝑇2o 𝑇1
𝑇2
𝑇1
>
>
5. 91
(𝑇2o 𝑇1)(𝑥;𝑦) = 𝑇2[𝑇1(𝑥; 𝑦)] = 𝑇2[(𝑥;𝑥 − 𝑦;𝑦)] = (𝑥 + 𝑦; 𝑥 − 𝑦)
Por lo tanto, (𝑇2o 𝑇1)(𝑥;𝑦) = (𝑥 + 𝑦;𝑥 − 𝑦)
b) Apliquemos 𝑇2o 𝑇1 al vector 𝑢 = (3;1)
Siendo, (𝑇2o 𝑇1)(𝑥;𝑦) = (𝑥 + 𝑦;𝑥 − 𝑦) (𝑇2o 𝑇1)(3; 1) = (4;2)
c) Apliquemos 𝑇2o 𝑇1 al subespacio vectorial 𝑆 = {(𝑎;𝑏) ∈ 𝑅2 𝑏 = 2𝑎
⁄ }
Si 𝑆 = {(𝑎;𝑏) ∈ 𝑅2 𝑏 = 2𝑎
⁄ } 𝑆 = {(𝑎;2𝑎) ∈ 𝑅2 𝑎 ∈ 𝑅
⁄ }, es una recta.
Si 𝑢 = (𝑎;𝑏) ∈ 𝑆, 𝑢 = (𝑎;2𝑎) (𝑇2o 𝑇1)(𝑎;2𝑎) = (3𝑎; −𝑎) = 𝑎(3; −1)
𝑇(𝑆) = {(3𝑎;−𝑎) ∈ 𝑅2 𝑎 ∈ 𝑅
⁄ } , también es una recta.
d) Obteniendo las matrices asociadas correspondientes respecto a las bases canónicas:
i) Para: 𝑇1 (𝑥:𝑦) = (𝑥; 𝑥 − 𝑦;𝑦) o 𝑇1 [
𝑥
𝑦] = [
𝑥
𝑥 − 𝑦
𝑦
] = [
𝑥 + 0𝑦
𝑥 − 𝑦
0𝑥 + 𝑦
] = [
1 0
1 −1
0 1
] [
𝑥
𝑦]
Su matriz asociada es: 𝐴𝑇
1
= [
1 0
1 −1
0 1
]
3×2
ii) Para: 𝑇2(𝑥;𝑦; 𝑧) = (𝑥 + 𝑧; 𝑦) o 𝑇2 [
𝑥
𝑦
𝑧
] = [
𝑥 + 𝑧
𝑦 ] = [
𝑥 + 0𝑦 + 𝑧
0𝑥 + 𝑦 + 0𝑧
]=[
1 0 1
0 1 0
][
𝑥
𝑦
𝑧
]
Su matriz asociada es: 𝐴𝑇
2
= [
1 0 1
0 1 0
]
2×3
iii) Para la composición: (𝑇2o 𝑇1)(𝑥; 𝑦) = (𝑥 + 𝑦;𝑥 − 𝑦)
(𝑇2o 𝑇1) [
𝑥
𝑦] = [
𝑥 + 𝑦
𝑥 − 𝑦] = [
1 1
1 −1
][
𝑥
𝑦]
Su matriz asociada es: 𝐴𝑇
2𝑜𝑇1
= [
1 1
1 −1
]
2×2
Otra forma de hallar 𝐴 de la transformación lineal composición es dada por el producto de
matrices asociadas de cada transformación lineal que forman la composición, según una
proposición anterior:
7. 93
o (𝑇1o 𝑇2) [
𝑥
𝑦
𝑧
] = [
𝑥 + 𝑧
𝑥 + 𝑧 − 𝑦
𝑦
] = [
𝑥 + 0𝑦 + 𝑧
𝑥 − 𝑦 + 𝑧
0𝑥 + 𝑦 + 0𝑧
] = [
1 0 1
1 −1 1
0 1 0
][
𝑥
𝑦
𝑧
]
Su matriz asociada es: 𝐴𝑇
1𝑜𝑇2
= [
1 0 1
1 −1 1
0 1 0
]
3×3
Otraforma dehallar la matriz asociada de𝑇1𝑜𝑇2 , es dada porel productode matrices asociadas
de las transformaciones lineales, usando una proposición anterior:
𝐴𝑇
1𝑜𝑇2
= 𝐴𝑇
1
𝐴𝑇2
= [
1 0
1 −1
0 1
]
3×2
.[
1 0 1
0 1 0
]
2×3
= [
1 0 1
1 −1 1
0 1 0
]
3×3
c) Aplicando 𝑇1o 𝑇2 al vector 𝑢 = (3; −4;2)
Aplicando (𝑇1 o 𝑇2)(𝑥; 𝑦;𝑧) = (𝑥 + 𝑧; 𝑥 + 𝑧 − 𝑦;𝑦) al vector 𝑢 = (3; −4;2)
(𝑇1o 𝑇2)(3;−4; 2) =(5; 9;−4)
d) Aplicando 𝑇1o 𝑇2 al subespacio vectorial 𝑆 = {(𝑥;𝑦;𝑧) ∈ 𝑅3 (𝑥;𝑦;𝑧) = 𝑡(3;2; 1),𝑡 ∈ 𝑅
⁄ }
Aplicando (𝑇1o 𝑇2)(𝑥; 𝑦;𝑧) = (𝑥 + 𝑧; 𝑥 + 𝑧 − 𝑦;𝑦) al subespacio vectorial 𝑆 =
{(𝑥;𝑦;𝑧) ∈ 𝑅3 (𝑥;𝑦;𝑧) = 𝑡(3;2; 1),𝑡 ∈ 𝑅
⁄ }, es una recta.
Si 𝑢 = (3𝑡;2𝑡; 𝑡) ∈ 𝑆 (𝑇1 o 𝑇2)(3𝑡; 2𝑡;𝑡) = (4𝑡;2𝑡; 2𝑡) = 𝑡(4;2;2)
𝑇(𝑆) = {(𝑥;𝑦;𝑧) ∈ 𝑅3 (𝑥;𝑦;𝑧) = 𝑡(4;2; 2),𝑡 ∈ 𝑅
⁄ }
Que también es un subespacio vectorial (una recta)
EJEMPLO 3. Sean 𝑇1:𝑅3 → 𝑅 tal que 𝑇1(𝑥;𝑦;𝑧) = 𝑥 − 𝑦 + 𝑧, y 𝑇2:𝑅 → 𝑅2 tal que 𝑇2(𝑥) =
(𝑥; −2𝑥).
a) Hallar 𝑇2o 𝑇1
b) Hallar las matrices asociadas
c) ¿Es posible hallar 𝑇1 o 𝑇2? Justifique su respuesta.
Solución.
a) La transformación lineal composición 𝑇2o 𝑇1 es posible como se ve en seguida:
(𝑇2o 𝑇1)(𝑥;𝑦;𝑧) =𝑇2[𝑇1(𝑥;𝑦;𝑧)] =𝑇2[(𝑥 − 𝑦 + 𝑧)] = (𝑥 − 𝑦 + 𝑧; −2𝑥 + 2𝑦 − 2𝑧)
Por lo tanto, (𝑇2o 𝑇1)(𝑥;𝑦;𝑧) = (𝑥 − 𝑦 + 𝑧; −2𝑥 + 2𝑦 − 2𝑧)
8. 94
b) Obteniendo las matrices asociadas correspondientes:
i) Para: 𝑇1 (𝑥;𝑦;𝑧) = 𝑥 − 𝑦 + 𝑧 o 𝑇1 [
𝑥
𝑦
𝑧
] = [𝑥 − 𝑦 + 𝑧] = [1 −1 1] [
𝑥
𝑦
𝑧
]
Su matriz asociada es: 𝐴𝑇
1
= [1 −1 1]1×3
ii) Para: 𝑇2(𝑥) = (𝑥;−2𝑥) o 𝑇2[𝑥] = [
𝑥
−2𝑥
] = [
1
−2
] [𝑥]
Su matriz asociada es: 𝐴𝑇
2
= [
1
−2
]
2×1
iii) Para: (𝑇2o 𝑇1)(𝑥; 𝑦;𝑧) = (𝑥 − 𝑦 + 𝑧; −2𝑥 + 2𝑦 − 2𝑧)
O también: (𝑇2o 𝑇1) [
𝑥
𝑦
𝑧
] = [
𝑥 − 𝑦 + 𝑧
−2𝑥 + 2𝑦 − 2𝑧
] = [
1 −1 1
−2 2 −2
] [
𝑥
𝑦
𝑧
]
Su matriz asociada es: 𝐴𝑇
2o 𝑇1
= [
1 −1 1
−2 2 −2
]
2×3
Otra forma de hallar la matriz asociada es como el producto de las transformaciones lineales:
𝐴𝑇
2o 𝑇1
= 𝐴𝑇
2
𝐴𝑇1
= [
1
−2
]
2×1
[1 −1 1]1×3 = [
1 −1 1
−2 2 −2
]
2×3
c) La composición 𝑇1o 𝑇2 no es posible. ¿Por qué?
Sean 𝑇1:𝑅3 → 𝑅 tal que 𝑇1(𝑥;𝑦; 𝑧) = 𝑥 − 𝑦 + 𝑧, y 𝑇2: 𝑅 → 𝑅2 tal que 𝑇2(𝑥) = (𝑥; −2𝑥). ¿Es
posible hallar 𝑇1o 𝑇2?
No es posible, porque la dimensión de llegada de 𝑇1 es diferente de la dimensión del espacio de
partida de 𝑇2.
Probando esto: (𝑇1o 𝑇2)(𝑥) = 𝑇1[𝑇2(𝑥)] = 𝑇1(𝑥;−2𝑥), ¡aquí no es posible aplicar 𝑇1 dado que
esta aplica a vectores de tres componentes!
EJEMPLO 4. Sean 𝑇1:𝑅3 → 𝑃≤1 tal que 𝑇1(𝑎;𝑏;𝑐) = 𝑎 + (𝑏 + 𝑐)𝑥, y 𝑇2:𝑃≤1 → 𝑀2×2 tal que
𝑇2(𝑚 + 𝑛𝑥) = [
𝑚 𝑚 + 𝑛
𝑚 − 𝑛 𝑛
]
a) Hallar 𝑇2o 𝑇1
b) Hallar las matrices asociadas, respecto a las bases canónicas.
9. 95
c) ¿Es posible hallar 𝑇1 o 𝑇2? Justifique su respuesta.
Solución.
a) Hallando: 𝑇2o 𝑇1
(𝑇2o 𝑇1) = 𝑇2[𝑇1(𝑎; 𝑏;𝑐)] = 𝑇2[𝑎 + (𝑏 + 𝑐)𝑥] = [
𝑎 𝑎 + 𝑏 + 𝑐
𝑎 − 𝑏 − 𝑐 𝑏 + 𝑐
]
Por lo tanto, (𝑇2o 𝑇1) =[
𝑎 𝑎 + 𝑏 + 𝑐
𝑎 − 𝑏 − 𝑐 𝑏 + 𝑐
]
b) Hallar las matrices asociadas, respecto a las bases canónicas.
Base de 𝑅3, [𝑣] = {(1;0; 0);(0; 1;0); (0;0;1)}, de 𝑃≤1 es [𝑤] = {1;𝑥} y de 𝑀2×2 es [𝑤] =
{[
1 0
0 0
] , [
0 1
0 0
] ,[
0 0
1 0
], [
0 0
0 1
]}
i) Siendo: 𝑇1:𝑅3 → 𝑃≤1 tal que 𝑇1(𝑎; 𝑏;𝑐) = 𝑎 + (𝑏 + 𝑐)𝑥 o 𝑇1 [
𝑎
𝑏
𝑐
] = 𝑎 + (𝑏 + 𝑐)𝑥
La matriz asociada es de la forma: 𝐴𝑇1
= [
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
]
2×3
𝑇1 [
1
0
0
]= 1 + 0𝑥 = 𝑎11(1) + 𝑎21(𝑥)= 1(1) + 0(𝑥) 𝑇1 [
1
0
0
]
[𝑤]
= [
1
0
]
𝑇1 [
0
1
0
]= 0 + 𝑥 = 𝑎12(1) + 𝑎22(𝑥)= 0(1) + 1(𝑥) 𝑇1 [
0
1
0
]
[𝑤]
= [
0
1
]
𝑇1 [
0
0
1
]= 0 + 𝑥 = 𝑎13(1) + 𝑎23(𝑥)= 0(1) + 1(𝑥) 𝑇1 [
0
0
1
]
[𝑤]
= [
0
1
]
Por lo tanto, la matriz asociada a 𝑇1 es: 𝐴𝑇
1
= [
1 0 0
0 1 1
]
3×2
ii) Siendo: 𝑇2:𝑃≤1 → 𝑀2×2 tal que 𝑇2(𝑚 + 𝑛𝑥) = [
𝑚 𝑚 + 𝑛
𝑚 − 𝑛 𝑛
] ; [𝑤] = {1;𝑥}
La matriz asociada es de la forma: 𝐴𝑇2
= [
𝑎11 𝑎12
𝑎21 𝑎22
𝑎13 𝑎14
𝑎23 𝑎24
]
4×2
𝑇2(1) =[
1 1
1 0
] = 𝑐1 [
1 0
0 0
] + 𝑐2[
0 1
0 0
] + 𝑐3 [
0 0
1 0
] + 𝑐4 [
0 0
0 1
]
10. 96
𝑇2(1) =[
1 1
1 0
] = 1 [
1 0
0 0
] + 1[
0 1
0 0
] + 1 [
0 0
1 0
] + 0 [
0 0
0 1
] 𝑇2(1)[𝑢] = [
1
1
1
0
]
𝑇2(1) =[
1 1
1 0
] = 𝑐1 [
1 0
0 0
] + 𝑐2[
0 1
0 0
] + 𝑐3 [
0 0
1 0
] + 𝑐4 [
0 0
0 1
]
𝑇2(𝑥) =[
0 1
−1 1
] = 0 [
1 0
0 0
] + 1[
0 1
0 0
] + (−1)[
0 0
1 0
] + 1 [
0 0
0 1
] 𝑇2(1)[𝑢] = [
0
1
−1
1
]
Por lo tanto, la matriz asociada a 𝑇2 es: 𝐴𝑇
2
= [
1 0
1 1
1 −1
0 1
]
4×2
iii) La matriz asociada de la composición: 𝑇2𝑜𝑇1: 𝑅3 → 𝑀2×2
(𝑇2o 𝑇1) = 𝑇2[𝑇1(𝑎;𝑏; 𝑐)] = 𝑇2[𝑎 + (𝑏 + 𝑐)𝑥] = [
𝑎 𝑎 + 𝑏 + 𝑐
𝑎 − 𝑏 − 𝑐 𝑏 + 𝑐
]
Usando la proposiciónes:𝐴𝑇
2𝑜𝑇1
= 𝐴𝑇
2
𝐴𝑇
1
=[
1 0
1 1
1 −1
0 1
]
4×2
[
1 0 0
0 1 1
]
2×3
=[
1
1
0
1
0
1
1
0
−1
1
−1
1
]
4×3
Por lo tanto, la matriz asociada a 𝑇2𝑜𝑇1 es: 𝐴𝑇
2𝑜𝑇1
= [
1
1
0
1
0
1
1
0
−1
1
−1
1
]
4×3
c) ¿Es posible hallar 𝑇1 o 𝑇2? Justifique su respuesta.
No es posible, porque la dimensión de llegada de 𝑇1 , es 𝑛 = 2, es diferente de la dimensión del
espacio de partida de 𝑇2 , 𝑚 = 4.
Probando esto: (𝑇1o 𝑇2)(𝑚 + 𝑛𝑥) = 𝑇1[𝑇2(𝑚 + 𝑛𝑥)] = 𝑇1 [
𝑚 𝑚 + 𝑛
𝑚 − 𝑛 𝑛
],
¡Aquí no es posible aplicar 𝑇1 dado que esta aplica a vectores de la forma (𝑎;𝑏; 𝑐) ∈ 𝑅3!
3.4 TRANSFORMACIONES LINEALES INVERTIBLES.
DEFINICIÓN. Una transformación lineal 𝑇:𝑉 → 𝑊 siendo (𝑉;𝐾,+; ∙), (𝑊;𝐾,+; ∙) es invertible
si existe una aplicación𝐹:𝑊 → 𝑉 tal que la composición𝑇o 𝐹 = 𝐼𝑊 y la composición 𝐹𝑜𝑇 = 𝐼𝑉.
Asumiendo que 𝐹 es la inversa, se tiene 𝐹 = 𝑇−1 entonces 𝑇o𝐹 = 𝑇𝑜𝑇−1 = 𝐼𝑊 es la aplicación
identidad en 𝑊 y 𝐹o𝑇 = 𝑇−1𝑜𝑇 = 𝐼𝑉 es la aplicación identidad en 𝑉 .
OBSERVACIÓN. Si la transformación lineal es invertible, entonces la inversa es única y en
adelante se denotará por 𝑇−1.
11. 97
Además: 𝑇o 𝑇−1 = 𝐼𝑊 es la identidad en 𝑊 ; y, 𝑇−1o T = 𝐼𝑉 es la identidad en 𝑉.
LEMA(PROPIEDAD).Si𝑇:𝑉 → 𝑊,siendo (𝑉;𝐾;+; ∙), (𝑊;𝐾;+; ∙), es una transformación lineal
invertible, su inversa 𝑇−1:𝑊 → 𝑉 también es una transformación lineal.
Demostración.
Siendo 𝑘1; 𝑘2 ∈ 𝐾 y 𝑌1; 𝑌2 ∈ 𝑊 𝑇−1(𝑘1𝑌
1 + 𝑘2𝑌
2) = 𝑘1𝑇−1(𝑌
1)+ 𝑘2𝑇−1(𝑌2) ¡Probar!
En efecto:
Recordar: Si 𝑇 es invertible 𝑇 es inyectiva y sobreyectiva.
Si 𝑌1; 𝑌2 ∈ 𝑊 ∃ 𝑋1; 𝑋2 ∈ 𝑉 tal que 𝑇(𝑋1) = 𝑌1y 𝑇(𝑋2) = 𝑌2por ser sobreyectiva
También se tiene que: 𝑋1 = 𝑇−1(𝑌
1)y 𝑋2 = 𝑇−1(𝑌
2), pues 𝑇 es invertible.
Como 𝑉 es un espacio vectorial: 𝑘1𝑋1 + 𝑘2𝑋2 ∈ 𝑉
𝑇(𝑘1𝑋1 + 𝑘2𝑋2) = 𝑘1𝑇(𝑋1) + 𝑘2𝑇(𝑋2)= 𝑘1𝑌1 + 𝑘2𝑌2
𝑘1𝑋1 + 𝑘2𝑋2 es único en 𝑉 que es aplicado en 𝑘1𝑌
1 + 𝑘2𝑌2 ∈ 𝑊 es único en 𝑊.
𝑇−1(𝑘1𝑌
1 + 𝑘2𝑌
2)= 𝑘1𝑋1 + 𝑘2𝑋2 = 𝑘1𝑇−1(𝑌
1)+ 𝑘2𝑇−1(𝑌
2)
Por lo tanto, 𝑇−1 es una transformación lineal.
OBSERVACIÓN(PROPIEDAD).Siendo𝑇:𝑉 → 𝑊 una transformación lineal, siendo (𝑉;𝐾,+; ∙)y
(𝑊;𝐾,+; ∙) espacios vectoriales. En general, sea 𝑇 ∈ 𝐿(𝑉;𝑊).
a) 𝑇 es invertible 𝑇 es biyectiva.
b) 𝑇 es inyectiva El núcleo de 𝑇, 𝑁(𝑇) = {0𝑉}
c) 𝑇 es sobreyectiva 𝑇(𝑉) = 𝑊
Prueba.
a1) () Si 𝑇 es invertible 𝑇 es biyectiva.
i) Si 𝑇 es invertible 𝑇 es inyectiva.
𝑇
(𝑉; 𝐾, +; ∙) (𝑊;𝐾,+; ∙)
𝑋 . . 𝑇(𝑋)
>
<
𝑇−1
12. 98
Sean 𝑣𝑖;𝑣𝑗 ∈ 𝑉 y suponga que 𝑇(𝑣𝑖) = 𝑇(𝑣𝑗) 𝑇−1[𝑇(𝑣𝑖)] = 𝑇−1[𝑇(𝑣𝑗)]
(𝑇−1𝑜𝑇)(𝑣𝑖) = (𝑇−1𝑜𝑇)(𝑣𝑗) 𝑣𝑖 = 𝑣𝑗
Por lo tanto, 𝑇 es inyectiva.
ii) Si 𝑇 es invertible 𝑇 es sobreyectiva.
Sea 𝑇 ∈ 𝐿(𝑉;𝑊). ∀ 𝑤 ∈ 𝑊, ∃ 𝑣 ∈ 𝑉 tal que 𝑇(𝑣) = 𝑤
Como 𝑇 es invertible ∀ 𝑤 ∈ 𝑊, ∃ 𝑣 ∈ 𝑉 tal que 𝑇−1(𝑤) = 𝑣 𝑇[𝑇−1(𝑤)] = 𝑇(𝑣) 𝑇(𝑣) = 𝑤
Por lo tanto, 𝑇 es sobreyectiva.
a2) () Si 𝑇 es biyectiva 𝑇 es invertible.
Se debe probar que: ∃ 𝑇−1:𝑊 → 𝑉 tal que 𝑇−1𝑜𝑇 = 𝐼𝑑𝑉 𝑇𝑜𝑇−1 = 𝐼𝑑𝑊
Por definición de 𝑇, ∀ 𝑣 ∈ 𝑉 𝑇(𝑣) = 𝑤 ∈ 𝑊 (1)
Por ser 𝑇 sobreyectiva, ∀ 𝑤 ∈ 𝑊, ∃ 𝑣 ∈ 𝑉 tal que 𝑇
̅(𝑤) = 𝑣 (2)
Por ser 𝑇 inyetiva el 𝑣 es único. Si 𝑇(𝑢) = 𝑇(𝑣) 𝑢 = 𝑣
De (1), 𝑇(𝑣) = 𝑤
𝑇
̅[𝑇(𝑣)] = 𝑇
̅(𝑤) 𝑇
̅[𝑇(𝑣)] = 𝑣 (𝑇
̅𝑜𝑇)(𝑣) = 𝑣 𝑇
̅𝑜𝑇 = 𝐼𝑑𝑉 (3)
De (2), 𝑇
̅(𝑤) = 𝑣
𝑇[𝑇
̅(𝑤)] = 𝑇(𝑣) 𝑇[𝑇
̅(𝑤)] = 𝑤 (𝑇𝑜𝑇
̅)(𝑤) = 𝑤 𝑇𝑜𝑇
̅ = 𝐼𝑑𝑊 (4)
Es decir 𝑇
̅ = 𝑇−1 es la inversa de 𝑇.
Por lo tanto, 𝑇 es invertible.
EJEMPLO 1. Sea la aplicación 𝑇: 𝑅3 → 𝑅3 tal que 𝑇(𝑥;𝑦; 𝑧) = (2𝑦;𝑦 − 𝑥; 𝑦 + 𝑧). Verificar las
afirmaciones que se dan:
a) La aplicación 𝑇 es una transformación lineal.
b) La transformación lineal 𝑇 es invertible.
c) Si 𝑇 es invertible, hallar la transformación lineal inversa 𝑇−1:𝑅3 → 𝑅3.
d) Compruebe que: 𝑇−1o 𝑇 = 𝐼𝑅3.
e) Aplique 𝑇 al subespacio vectorial 𝑆 = {(𝑥;𝑦,𝑧) ∈ 𝑅3 (𝑥;𝑦;𝑧) = 𝑡(1;2;3)
⁄ ,𝑡 ∈ 𝑅} es una
recta.
13. 99
f) Aplique 𝑇−1 al subespacio vectorial 𝑁 = {(𝑎;𝑏; 𝑐) ∈ 𝑅3 2𝑎 − 3𝑏 + 𝑐 = 0
⁄ } es un plano.
Solución.
a) La aplicación 𝑇 es una transformación lineal.
En efecto: 𝑇(𝑥; 𝑦;𝑧) = (2𝑦;𝑦 − 𝑥;𝑦 + 𝑧)
𝑇 𝑘1(𝑥1;𝑦1;𝑧1) + 𝑘2(𝑥2;𝑦2;𝑧2) = 𝑇(𝑘1𝑥1 + 𝑘2𝑥2;𝑘1𝑦1 + 𝑘2𝑦2;𝑘1𝑧1 + 𝑘2𝑧2)
= (2𝑘1𝑦1 + 2𝑘2𝑦2;𝑘1𝑦1 + 𝑘2𝑦2 − 𝑘1𝑥1 − 𝑘2𝑥2;𝑘1𝑦1 + 𝑘2𝑦2 + 𝑘1𝑧1 + 𝑘2𝑧2)
= (2𝑘1𝑦1;𝑘1𝑦1 − 𝑘1𝑥1;𝑘1𝑦1 + 𝑘1𝑧1) + (2𝑘2𝑦2;𝑘2𝑦2 − 𝑘2𝑥2;𝑘2𝑦2 + 𝑘2𝑧2)
= 𝑘1(2𝑦1;𝑦1 − 𝑥1;𝑦1 + 𝑧1) + 𝑘2(2𝑦2;𝑦2 − 𝑥2;𝑦2 + 𝑧2)
= 𝑘1𝑇(𝑥1;𝑦1; 𝑧1) + 𝑘2𝑇(𝑥2;𝑦2;𝑧2)
Por lo tanto, 𝑇 es una transformación lineal.
b) La transformación lineal 𝑇 es invertible. (𝑇 es inyectiva y sobreyectiva)
Se debe probar que 𝑁(𝑇) = {0𝑉} y que 𝑇 es sobreyectiva.
i) ∀ (𝑥;𝑦;𝑧) ∈ 𝑁(𝑇) 𝑇(𝑥;𝑦; 𝑧) = (0; 0;0) (2𝑦;𝑦 − 𝑥; 𝑦 + 𝑧) = (0; 0;0)
{
2𝑦 = 0
𝑦 − 𝑥 = 0
𝑦 + 𝑧 = 0
{
𝑥 = 0
𝑦 = 0
𝑧 = 0
Con lo cual 𝑁(𝑇) = {(0;0; 0)} 𝑇 es inyectiva.
ii) Si 𝑁(𝑇) = {(0; 0;0)} 𝐷𝑖𝑚 𝑁(𝑇) = 0
Como: 𝐷𝑖𝑚 𝑁(𝑇) + 𝐷𝑖𝑚 𝐼𝑚(𝑇) = 𝐷𝑖𝑚(𝑅3) = 3 𝐷𝑖𝑚 𝐼𝑚(𝑇) = 3 ; 𝐼𝑚(𝑇) = 𝑅3
𝑇 es sobreyectiva.
De (i) y (ii) la transformación lineal 𝑇 es invertible.
c) Si 𝑇 es invertible, hallar la transformación lineal inversa 𝑇−1:𝑅3 → 𝑅3.
Sea (𝑎; 𝑏;𝑐) ∈ 𝑅3 en el conjunto de llegada ∃ (𝑥;𝑦; 𝑧) ∈ 𝑅3 en el conjunto partida, tal que
𝑇(𝑥; 𝑦;𝑧) = (𝑎;𝑏;𝑐) y que 𝑇−1(𝑎;𝑏; 𝑐) = (𝑥; 𝑦:𝑧) (2𝑦;𝑦 − 𝑥; 𝑦 + 𝑧) = (𝑎;𝑏; 𝑐)
{
2𝑦 = 𝑎
𝑦 − 𝑥 = 𝑏
𝑦 + 𝑧 = 𝑐
{
𝑥 =
𝑎
2
− 𝑏
𝑦 =
𝑎
2
𝑧 = 𝑐 −
𝑎
2
14. 100
Por lo tanto, la transformación lineal inversa será: 𝑇−1(𝑎;𝑏; 𝑐) = (
𝑎
2
− 𝑏;
𝑎
2
;𝑐 −
𝑎
2
)
d) Compruebe que: 𝑇−1o 𝑇 = 𝐼𝑅3. También 𝑇𝑜𝑇−1 = 𝐼𝑅3
En efecto: (𝑇−1o 𝑇)(𝑥;𝑦; 𝑧) = 𝑇−1[𝑇(𝑥;𝑦;𝑧)] = 𝑇−1(2𝑦;𝑦 − 𝑥;𝑦 + 𝑧)
= (
2𝑦
2
− 𝑦 + 𝑥;
2𝑦
2
;𝑦 + 𝑧 −
2𝑦
2
) (𝑇−1o𝑇)(𝑥;𝑦;𝑧) = (𝑥;𝑦,𝑧) = 𝐼𝑅3
Por lo tanto: (𝑇−1o𝑇) = 𝐼𝑅3
e) Aplique 𝑇 al subespacio vectorial 𝑆 = {(𝑥;𝑦,𝑧) ∈ 𝑅3 (𝑥;𝑦;𝑧) = 𝑡(1;2;3)
⁄ ,𝑡 ∈ 𝑅} es una
recta.
𝑇(𝑥; 𝑦;𝑧) = 𝑇(𝑡; 2𝑡;3𝑡) = (2(2𝑡);2𝑡 − 𝑡;2𝑡 + 3𝑡) = (4𝑡;𝑡; 5𝑡) = 𝑡(4;1; 5)
En el espacio de llegada 𝑇(𝑆) = {(𝑥;𝑦, 𝑧) ∈ 𝑅3 (𝑥;𝑦; 𝑧) = 𝑡(4; 1;5)
⁄ ,𝑡 ∈ 𝑅}es una recta.
f) Aplique 𝑇−1 al subespacio vectorial 𝑁 = {(𝑎;𝑏; 𝑐) ∈ 𝑅3 2𝑎 − 3𝑏 + 𝑐 = 0
⁄ } es un plano.
𝑇−1(𝑎;𝑏; 𝑐) = 𝑇−1(𝑎;𝑏; 3𝑏 − 2𝑎) = (
𝑎
2
− 𝑏;
𝑎
2
; 3𝑏 − 2𝑎 −
𝑎
2
) = (
𝑎
2
− 𝑏;
𝑎
2
;3𝑏 −
5𝑎
2
) = (𝑥;𝑦; 𝑧)
En el conjunto de llegada (
𝑎
2
− 𝑏;
𝑎
2
;3𝑏 −
5𝑎
2
) = (𝑥;𝑦;𝑧)
{
𝑎
2
− 𝑏 = 𝑥
𝑎
2
= 𝑦
3𝑏 −
5𝑎
2
= 𝑧
{
𝑥 − 𝑦 = −𝑏
𝑧 = 3𝑏 −
5𝑎
2
𝑧 = −3𝑥 − 2𝑦; es la ecuación de un plano.
En el espacio de llegada 𝑇−1(𝑁) = {(𝑥;𝑦; 𝑧) ∈ 𝑅3 𝑧 = −3𝑥 − 2𝑦
⁄ }, también es un plano.
EJEMPLO 2. Sea la transformación lineal 𝑇: 𝑅2 → 𝑃≤1tal que 𝑇(𝑎; 𝑏) = 2𝑎 + (𝑎 + 𝑏)𝑥
a) Muestre que 𝑇 es una transformación lineal.
b) Muestre que 𝑇 es biyectiva.
c) Hallar la transformación lineal inversa de 𝑇.
d) Muestre que 𝑇−1:𝑃≤1 → 𝑅2 es una transformación lineal.
e) Hallar las matrices asociadas a la transformaciones lineales 𝑇 y 𝑇−1.