Consider T : R2 R4 defined by T (x) = Ax, where A = 1 2 2 4 4 8 8 16 . For each x below, find T (x) and thereby determine whether x is in Ker(T ). (a) x = (10, 5). (b) x = (1, 1). (c) x = (2, 1).
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Table of Contents
1.1
Differential Equations Everywhere
1.10
Numerical Solution to First-Order Differential Equations
1.11
Some Higher-Order Differential Equations
1.12
Basic Theory of Differential Equations
1.2
Basic Ideas and Terminology
1.3
The Geometry of First-Order Differential Equations
1.4
Separable Differential Equations
1.5
Some Simple Population Models
1.6
First-Order Linear Differential Equations
1.7
Modeling Problems Using First-Order Linear Differential Equations
1.8
Change of Variables
1.9
Exact Differential Equations
2.1
Matrices: Definitions and Notation
2.2
Matrix Algebra
2.3
Terminology for Systems of Linear Equations
2.4
Row-Echelon Matrices and Elementary Row Operations
2.5
Gaussian Elimination
2.6
The Inverse of a Square Matrix
2.7
Elementary Matrices and the LU Factorization
2.8
The Invertible Matrix Theorem I I
2.9
Chapter Review
3.1
The Definition of the Determinant
3.2
Properties of Determinants F
3.3
Cofactor Expansions
3.4
Summary of Determinants
3.5
Chapter Review
4
Vector Spaces
4.1
Vectors in Rn
4.2
Definition of a Vector Space
4.3
Subspaces
4.4
Spanning Sets
4.5
Linear Dependence and Linear Independence
4.6
Bases and Dimension
4.7
Change of Basis
4.8
Row Space and Column Space
4.9
The Rank-Nullity Theorem
5.1
Definition of an Inner Product Space
5.2
Orthogonal Sets of Vectors and Orthogonal Projections
5.4
Least Squares Approximation
5.5
Inner Product Spaces
6.1
Definition of a Linear Transformation
6.2
Transformations of R2
6.3
The Kernel and Range of a Linear Transformation
6.4
Additional Properties of Linear Transformations
6.5
The Matrix of a Linear Transformation
6.6
Linear Transformations
7.1
The Eigenvalue/Eigenvector Problem
7.2
General Results for Eigenvalues and Eigenvectors
7.3
Diagonalization
7.4
An Introduction to the Matrix Exponential Function
7.5
Orthogonal Diagonalization and Quadratic Forms
7.6
Jordan Canonical Forms
7.7
The Algebraic Eigenvalue/Eigenvector Problem
8.1
General Theory for Linear Differential Equations
8.10
Linear Differential Equations of Order n
8.2
Constant Coefficient Homogeneous Linear Differential Equations
8.3
The Method of Undetermined Coefficients: Annihilators
8.4
Complex-Valued Trial Solutions
8.5
Oscillations of a Mechanical System
8.6
RLC Circuits
8.7
The Variation of Parameters Method
8.8
A Differential Equation with Nonconstant Coefficients
8.9
Reduction of Order
9.1
First-Order Linear Systems
9.10
Nonlinear Systems
9.11
Systems of Differential Equations
9.2
Vector Formulation
9.3
General Results for First-Order Linear Differential Systems
9.4
Vector Differential Equations: Nondefective Coefficient Matrix
9.5
Vector Differential Equations: Defective Coefficient Matrix
9.6
Variation-of-Parameters for Linear Systems
9.7
Some Applications of Linear Systems of Differential Equations
9.8
Matrix Exponential Function and Systems of Differential Equations
9.9
The Phase Plane for Linear Autonomous Systems
10.1
Definition of the Laplace Transform
10.10
The Laplace Transform and Some Elementary Applications
10.2
The Existence of the Laplace Transform and the Inverse Transform
10.3
Periodic Functions and the Laplace Transform
10.4
The Transform of Derivatives and Solution of Initial-Value Problems
10.5
The First Shifting Theorem
10.6
The Unit Step Function
10.7
The Second Shifting Theorem
10.8
Impulsive Driving Terms: The Dirac Delta Function
10.9
The Convolution Integral
11.1
Review of Power Series
11.2
Series Solutions about an Ordinary Point
11.3
The Legendre Equation
11.4
Series Solutions about a Regular Singular Point
11.5
Frobenius Theory
11.6
Bessels Equation of Order p
11.7
Series Solutions to Linear Differential Equations
Textbook Solutions for Differential Equations and Linear Algebra
Chapter 6.3 Problem 23
Question
Let T : V W and S : V W be linear transformations, and assume that {v1, v2,..., vk } spans V. Prove that if T (vi) = S(vi) for each i = 1, 2,..., k, then T = S; that is, T (v) = S(v) for each v V
Solution
The first step in solving 6.3 problem number 23 trying to solve the problem we have to refer to the textbook question: Let T : V W and S : V W be linear transformations, and assume that {v1, v2,..., vk } spans V. Prove that if T (vi) = S(vi) for each i = 1, 2,..., k, then T = S; that is, T (v) = S(v) for each v V
From the textbook chapter The Kernel and Range of a Linear Transformation you will find a few key concepts needed to solve this.
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full solution
Title
Differential Equations and Linear Algebra 4
Author
Stephen W. Goode
ISBN
9780321964670
Solved: Let T : V W and S : V W be linear transformations, and assume that {v1, v2,...
Chapter 6.3 textbook questions
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Chapter 6: Problem 1 Differential Equations and Linear Algebra 4
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Chapter 6: Problem 2 Differential Equations and Linear Algebra 4
Consider T : R3 R2 defined by T (x) = Ax, where A = 1 1 2 1 2 3 . For each x below, find T (x) and thereby determine whether x is in Ker(T ). (a) x = (7, 5, 1). (b) x = (21, 15, 2). (c) x = (35, 25, 5).
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Chapter 6: Problem 3 Differential Equations and Linear Algebra 4
For Problems 37, find Ker(T ) and Rng(T ), and give a geometrical description of each. Also, find dim[Ker(T )] and dim[Rng(T )], and verify Theorem 6.3.8.T : R2 R2 defined by T (x) = Ax, where A = 3 6 1 2 .
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Chapter 6: Problem 4 Differential Equations and Linear Algebra 4
For Problems 37, find Ker(T ) and Rng(T ), and give a geometrical description of each. Also, find dim[Ker(T )] and dim[Rng(T )], and verify Theorem 6.3.8.
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Chapter 6: Problem 5 Differential Equations and Linear Algebra 4
For Problems 37, find Ker(T ) and Rng(T ), and give a geometrical description of each. Also, find dim[Ker(T )] and dim[Rng(T )], and verify Theorem 6.3.8.T : R3 R3 defined by T (x) = Ax, where A = 1 2 1 2 3 1 5 8 1 .
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Chapter 6: Problem 6 Differential Equations and Linear Algebra 4
For Problems 37, find Ker(T ) and Rng(T ), and give a geometrical description of each. Also, find dim[Ker(T )] and dim[Rng(T )], and verify Theorem 6.3.8.. T : R3 R2 defined by T (x) = Ax, where A = 1 1 2 3 3 6 .
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Chapter 6: Problem 7 Differential Equations and Linear Algebra 4
For Problems 37, find Ker(T ) and Rng(T ), and give a geometrical description of each. Also, find dim[Ker(T )] and dim[Rng(T )], and verify Theorem 6.3.8.T : R3 R2 defined by T (x) = Ax, where A = 132 265 .
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Chapter 6: Problem 8 Differential Equations and Linear Algebra 4
For Problems 811, compute Ker(T ) and Rng(T ).The linear transformation T defined in Problem 27 in Section 6.1.
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Chapter 6: Problem 9 Differential Equations and Linear Algebra 4
For Problems 811, compute Ker(T ) and Rng(T ).The linear transformation T defined in Problem 28 in Section 6.1.
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Chapter 6: Problem 10 Differential Equations and Linear Algebra 4
For Problems 811, compute Ker(T ) and Rng(T ).
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Chapter 6: Problem 11 Differential Equations and Linear Algebra 4
For Problems 811, compute Ker(T ) and Rng(T ).The linear transformation T defined in Problem 30 in Section 6.1.
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Chapter 6: Problem 12 Differential Equations and Linear Algebra 4
Consider the linear transformation T : R3 R defined by T (v) = u, v, where u is a fixed nonzero vector in R3. (a) Find Ker(T ) and dim[Ker(T )], and interpret this geometrically. (b) Find Rng(T ) and dim[Rng(T )].
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Chapter 6: Problem 13 Differential Equations and Linear Algebra 4
Consider the linear transformation S : Mn(R) Mn(R) defined by S(A) = A + AT , where A is a fixed n n matrix. (a) Find Ker(S) and describe it. What is dim[Ker(S)]? (b) In the particular case when A is a 2 2 matrix, determine a basis for Ker(S), and hence, find its dimension.
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Chapter 6: Problem 14 Differential Equations and Linear Algebra 4
Consider the linear transformation T : Mn(R) Mn(R) defined by T (A) = AB B A, where B is a fixed n n matrix. Describe Ker(T ) in words.
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Chapter 6: Problem 15 Differential Equations and Linear Algebra 4
Consider the linear transformation T : P2(R) P2(R) defined by T (ax2 +bx +c) = ax2 +(a +2b +c)x +(3a 2b c), where a, b, and c are arbitrary constants.(a) Show that Ker(T ) consists of all polynomials of the form b(x 2), and hence, find its dimension. (b) Find Rng(T ) and its dimension.
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Chapter 6: Problem 16 Differential Equations and Linear Algebra 4
Consider the linear transformation T : P2(R) P1(R) defined by T (ax2 + bx + c) = (a + b) + (b c)x, where a, b, and c are arbitrary real numbers. Determine Ker(T ), Rng(T ), and their dimensions.
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Chapter 6: Problem 17 Differential Equations and Linear Algebra 4
Consider the linear transformation T : P1(R) P2(R) defined by T (ax + b) = (b a) + (2b 3a)x + bx2. Determine Ker(T ), Rng(T ), and their dimensions.
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Chapter 6: Problem 18 Differential Equations and Linear Algebra 4
Consider the linear transformation T : M2(R) P2(R) defined by T a b c d = (a b + d) + (a + b d)x2. Determine Ker(T ), Rng(T ), and their dimensions. 1
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Chapter 6: Problem 19 Differential Equations and Linear Algebra 4
Consider the linear transformation T : R2 M23(R) defined by T (x, y) = x y 0 2x + 2y 0 3x + 3y 9x 9y . Determine Ker(T ), Rng(T ), and their dimensions. 2
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Chapter 6: Problem 20 Differential Equations and Linear Algebra 4
Consider the linear transformation T : M24(R) M42(R) defined by T (A) = AT . Determine Ker(T ), Rng(T ), and their dimensions.
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Chapter 6: Problem 21 Differential Equations and Linear Algebra 4
Consider the linear transformation T : M24(R) M42(R) defined by T (A) = AT . Determine Ker(T ), Rng(T ), and their dimensions.
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Chapter 6: Problem 22 Differential Equations and Linear Algebra 4
(a) Let T : V W be a linear transformation, and suppose that dim[V] = n. If Ker(T ) = {0} and {v1, v2,..., vn} is any basis for V, prove that {T (v1), T (v2), . . . , T (vn)} is a basis for Rng(T ). (This fills in the missing details in the proof of Theorem 6.3.8.) (b) Show that the conclusion from part (a) fails to hold if Ker(T ) = {0}.
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Chapter 6: Problem 23 Differential Equations and Linear Algebra 4
Let T : V W and S : V W be linear transformations, and assume that {v1, v2,..., vk } spans V. Prove that if T (vi) = S(vi) for each i = 1, 2,..., k, then T = S; that is, T (v) = S(v) for each v V
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Chapter 6: Problem 24 Differential Equations and Linear Algebra 4
Let V be a vector space with basis{v1, v2,..., vk } and suppose T : V W is a linear transformation such that T (vi) = 0 for each i = 1, 2,..., k. Prove that T is the zero transformation; that is, T (v) = 0 for each v V.
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