Solutions for Concise Introduction to Linear Algebra 1st Edition by Hu (All Chapters included)

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