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Solutions for Concise Introduction to Linear Algebra 1st Edition by Hu (All Chapters included)
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Complete Solutions Manual for Concise Introduction to Linear Algebra 1st Edition by Qingwen Hu ; ISBN13: 9780367657703.....(Full Chapters included)...1. Vectors and linear systems 2. Solving linear systems 3. Vector spaces 4. Orthogonality 5. Determinants 6. Eigenvalues and eigenvectors 7. Si...

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  • October 26, 2024
  • 191
  • 2019/2020
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Solution Manual for Concise Introduction to
Linear Algebra

Qingwen Hu




Complete Chapter Solutions Manual
are included (Ch 1 to 9)




** Immediate Download
** Swift Response
** All Chapters included

,Contents



Preface ix

1 Vectors and linear systems 1

1.1 Vectors and linear combinations . . . . . . . . . . . . . . . . 1
1.2 Length, angle and dot product . . . . . . . . . . . . . . . . . 5
1.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Solving linear systems 11

2.1 Vectors and linear equations . . . . . . . . . . . . . . . . . . 11
2.2 Matrix operations . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Inverse matrices . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 LU decomposition . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Transpose and permutation . . . . . . . . . . . . . . . . . . . 25

3 Vector spaces 31

3.1 Spaces of vectors . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Nullspace, row space and column space . . . . . . . . . . . . 35
3.3 Solutions of Ax = b . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Rank of matrices . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Bases and dimensions of general vector spaces . . . . . . . . 45

4 Orthogonality 55

4.1 Orthogonality of the four subspaces . . . . . . . . . . . . . . 55
4.2 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Least squares approximations . . . . . . . . . . . . . . . . . . 72
4.4 Orthonormal bases and Gram–Schmidt . . . . . . . . . . . . 77

5 Determinants 85

5.1 Introduction to determinants . . . . . . . . . . . . . . . . . . 85
5.2 Properties of determinants . . . . . . . . . . . . . . . . . . . 89



vii

,viii Contents

6 Eigenvalues and eigenvectors 99

6.1 Introduction to eigenvectors and eigenvalues . . . . . . . . . 99
6.2 Diagonalizability . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3 Applications to differential equations . . . . . . . . . . . . . 111
6.4 Symmetric matrices and quadratic forms . . . . . . . . . . . 119
6.5 Positive definite matrices . . . . . . . . . . . . . . . . . . . . 134

7 Singular value decomposition 141

7.1 Singular value decomposition . . . . . . . . . . . . . . . . . . 141
7.2 Principal component analysis . . . . . . . . . . . . . . . . . . 149

8 Linear transformations 151

8.1 Linear transformation and matrix representation . . . . . . . 151
8.2 Range and null spaces of linear transformation . . . . . . . . 155
8.3 Invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . 158
8.4 Decomposition of vector spaces . . . . . . . . . . . . . . . . . 161
8.5 Jordan normal form . . . . . . . . . . . . . . . . . . . . . . . 164
8.6 Computation of Jordan normal form . . . . . . . . . . . . . . 165

9 Linear programming 173

9.1 Extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9.2 Simplex method . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.3 Simplex tableau . . . . . . . . . . . . . . . . . . . . . . . . . 176

, Chapter 1
Vectors and linear systems


1.1 Vectors and linear combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Length, angle and dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7




1.1 Vectors and linear combinations
Exercise 1.1.1.
[ ] [ ]
1 2
1. Let u = , v = . i) Sketch the directed line segments in R2 that
1 3
represents u and v, respectively; ii) Use the parallelogram law to visualize the
] 2u, 2u + 5v and 2v − 5u; iv) Solve the system
vector addition u + v; iii)[ Find
−1
of equations xu + yv = for (x, y) ∈ R2 and draw the row picture and
1
the column picture.
Solution: i)


v = (2, 3)




u = (1, 1)


(0, 0)



FIGURE 1.1: u = (1, 1), v = (2, 3)


ii)

1

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