Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Table of Contents
1.1
Vectors and Linear Combinations
1.2
Lengths and Dot Products
1.3
Matrices
2.1
Solving Linear Equations
2.2
The Idea of Elimination
2.3
Elimination Using Matrices
2.4
Rules for Matrix Operations
2.5
Inverse Matrices
2.6
Elimination = Factorization: A = L U
2.7
Transposes and Permutations
3.1
Spaces of Vectors
3.2
The Nullspace of A: Solving Ax = 0
3.3
The Rank and the Row Reduced Form
3.4
The Complete Solution to Ax = b
3.5
Independence, Basis and Dimension
3.6
Dimensions of the Four Subspaces
4.1
Orthogonality of the Four Subspaces
4.2
Projections
4.3
Least Squares Approximations
4.4
Orthogonal Bases and Gram-Schmidt
5.1
The Properties of Determinants
5.2
Permutations and Cofactors
5.3
Cramer's Rule, Inverses, and Volumes
6.1
Introduction to Eigenvalues
6.2
Diagonalizing a Matrix
6.3
Applications to Differential Equations
6.4
Symmetric Matrices
6.5
Positive Definite Matrices
6.6
Similar Matrices
6.7
Singular Value Decomposition (SVD)
7.1
The Idea of a Linear Transformation
7.2
The Matrix of a Linear Transformation
7.3
Diagonalization and the Pseudoinverse
8.1
Matrices in Engineering
8.2
Graphs and Networks
8.3
Markov Matrices, Population, and Economics
8.4
Linear Programming
8.5
Fourier Series: Linear Algebra for Functions
8.6
Linear Algebra for Statistics and Probability
8.7
Computer Graphics
9.1
Gaussian Elimination in Practice
9.2
Norms and Condition Numbers
9.3
Iterative Methods and Preconditioners
10.1
Complex Numbers
10.2
Hermitian and Unitary Matrices
10.3
The Fast Fourier Transform
Textbook Solutions for Introduction to Linear Algebra
Chapter 4.1 Problem 32
Question
Suppose I give you four nonzero vectors r, n, c, I in R 2 (a) What are the conditions for those to be bases for the four fundamental subspaces C(AT), N(A), C(A), N(AT) of a 2 by 2 matrix? (b) What is one possible matrix A
Solution
The first step in solving 4.1 problem number 32 trying to solve the problem we have to refer to the textbook question: Suppose I give you four nonzero vectors r, n, c, I in R 2 (a) What are the conditions for those to be bases for the four fundamental subspaces C(AT), N(A), C(A), N(AT) of a 2 by 2 matrix? (b) What is one possible matrix A
From the textbook chapter Orthogonality of the Four Subspaces you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Introduction to Linear Algebra 4
Author
Gilbert Strang
ISBN
9780980232714
Suppose I give you four nonzero vectors r, n, c, I in R 2 (a) What are the conditions
Chapter 4.1 textbook questions
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Chapter 4: Problem 1 Introduction to Linear Algebra 4
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Chapter 4: Problem 2 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 3 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 4 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 5 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 6 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 7 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 8 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 9 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 10 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 11 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 12 Introduction to Linear Algebra 4
Questions 1-12 grow out of Figures 4.2 and 4.3 with four subspaces.
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Chapter 4: Problem 13 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 14 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 15 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 16 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 17 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 18 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 19 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 20 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 21 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 22 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 23 Introduction to Linear Algebra 4
Questions 13-23 are about orthogonal subspaces.
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Chapter 4: Problem 24 Introduction to Linear Algebra 4
Questions 24-30 are about perpendicular columns and rows.
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Chapter 4: Problem 25 Introduction to Linear Algebra 4
Questions 24-30 are about perpendicular columns and rows.
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Chapter 4: Problem 26 Introduction to Linear Algebra 4
Questions 24-30 are about perpendicular columns and rows.
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Chapter 4: Problem 27 Introduction to Linear Algebra 4
Questions 24-30 are about perpendicular columns and rows.
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Chapter 4: Problem 28 Introduction to Linear Algebra 4
Questions 24-30 are about perpendicular columns and rows.
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Chapter 4: Problem 29 Introduction to Linear Algebra 4
Questions 24-30 are about perpendicular columns and rows.
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Chapter 4: Problem 30 Introduction to Linear Algebra 4
Questions 24-30 are about perpendicular columns and rows.
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Chapter 4: Problem 31 Introduction to Linear Algebra 4
The command N = nulI(A) will produce a basis for the nullspace of A. Then the command B = null(N') will produce a basis for the of A.
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Chapter 4: Problem 32 Introduction to Linear Algebra 4
Suppose I give you four nonzero vectors r, n, c, I in R 2 (a) What are the conditions for those to be bases for the four fundamental subspaces C(AT), N(A), C(A), N(AT) of a 2 by 2 matrix? (b) What is one possible matrix A
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Chapter 4: Problem 33 Introduction to Linear Algebra 4
Suppose I give you eight vectors r I, r2, nl, n2, CI, C2, 11,12 in R4. (a) What are the conditions for those pairs to be bases for the four fundamental subspaces of a 4 by 4 matrix? (b) What is one possible matrix A?
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