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Appendix B Answers and Hints for Selected Exercises
I Systems of Linear Equations
1 Introduction to Systems of Linear Equations
Exercises
1.
1.a
1.b
3.
5.
5.a
5.b
7.
7.a
7.b
7.c
7.d
9.
9.a
9.b
11. True/False Questions.
11.a True/False.
11.c True/False.
11.e True/False.
2 The Matrix Representation of a Linear System
Exercises
1.
1.a
1.b
1.c
3.
5.
6.
6.a True/False.
6.c True/False.
6.e True/False.
6.g True/False.
6.i True/False.
6.k True/False.
3 Row Echelon Forms
Exercises
1.
3.
5.
7.
7.a
7.b
9.
11.
12.
12.a True/False.
12.c True/False.
12.e True/False.
12.g True/False.
12.i True/False.
12.k True/False.
4 Vector Representation
Exercises
1.
3.
5.
5.a
5.b
5.c
7.
7.a
7.b
9.
11.
11.a True/False.
11.c True/False.
11.e True/False.
11.g True/False.
11.i True/False.
11.k True/False.
11.m True/False.
5 The Matrix-Vector Form of a Linear System
Exercises
1.
3.
5.
7.
7.a
7.b
9.
11.
13.
14.
14.a True/False.
14.c True/False.
14.e True/False.
14.g True/False.
14.i True/False.
14.k True/False.
6 Linear Dependence and Independence
Exercises
1.
3.
5.
5.a
5.b
7.
9.
9.a True/False.
9.c True/False.
9.e True/False.
9.g True/False.
9.i True/False.
9.k True/False.
7 Matrix Transformations
Exercises
1.
3.
5.
7.
9.
11.
12.
12.a True/False.
12.c True/False.
12.e True/False.
12.g True/False.
12.i True/False.
12.k True/False.
12.m True/False.
12.o True/False.
II Matrices
8 Matrix Operations
Exercises
1.
1.a
1.b
3.
3.a
3.b
3.c
5.
7.
9.
9.a
9.c
9.e
9.g
11.
11.a
11.c
13.
14.
14.a True/False.
14.c True/False.
14.e True/False.
14.g True/False.
9 Introduction to Eigenvalues and Eigenvectors
Exercises
1.
1.a
1.b
1.c
3.
3.a
3.b
3.c
3.d
5.
5.a
5.b
5.c
6.
6.a True/False.
6.c True/False.
6.e True/False.
6.g True/False.
6.i True/False.
6.k True/False.
10 The Inverse of a Matrix
Exercises
1.
2.
2.a
3.
3.a
3.b
3.c
5.
8.
8.a True/False.
8.c True/False.
8.e True/False.
8.g True/False.
8.i True/False.
11 The Invertible Matrix Theorem
Exercises
1.
3.
3.a True/False.
3.c True/False.
3.e True/False.
3.g True/False.
III The Vector Space \(\R^n\)
12 The Structure of \(\R^n\)
Exercises
1.
1.a
1.b
1.c
1.d
3.
5.
7.
7.a
7.a.i
7.a.ii
7.a.iii
7.a.iv
7.b
7.c
9.
11.
14.
14.a True/False.
14.c True/False.
14.e True/False.
14.g True/False.
14.i True/False.
14.k True/False.
13 The Null Space and Column Space of a Matrix
Exercises
1.
3.
5.
7.
8.
8.a True/False.
8.c True/False.
8.e True/False.
8.g True/False.
8.i True/False.
14 Eigenspaces of a Matrix
Exercises
1.
1.a
1.b
1.c
1.d
1.e
1.f
3.
5.
5.a
5.b
6.
6.a True/False.
6.c True/False.
6.e True/False.
6.g True/False.
15 Bases and Dimension
Exercises
1.
1.a
1.b
3.
3.a
3.b
3.c
3.d
5.
5.a
5.b
7.
8.
9.
11.
11.a
11.b
13.
13.a True/False.
13.c True/False.
13.e True/False.
13.g True/False.
13.i True/False.
13.k True/False.
16 Coordinate Vectors and Change of Basis
Exercises
1.
3.
5.
7.
7.a
7.b
9.
11.
11.a
11.b
11.c
13.
14.
14.a True/False.
14.c True/False.
14.e True/False.
14.g True/False.
14.i True/False.
14.k True/False.
IV Eigenvalues and Eigenvectors
17 The Determinant
Exercises
1.
3.
3.a
3.b
3.c
5.
7.
7.a True/False.
7.c True/False.
7.e True/False.
7.g True/False.
7.i True/False.
7.k True/False.
18 The Characteristic Equation
Exercises
1.
1.a
1.b
1.b.i
1.b.ii
1.b.iii
3.
5.
6.
6.a True/False.
6.c True/False.
6.e True/False.
6.g True/False.
6.i True/False.
19 Diagonalization
Exercises
1.
1.a
1.b
3.
5.
7.
7.a
7.b
9.
9.a
9.b
9.b.i
9.b.ii
11.
11.a
11.b
11.c
11.d
13.
13.a
13.b
15.
15.a True/False.
15.c True/False.
15.e True/False.
15.g True/False.
15.i True/False.
20 Approximating Eigenvalues and Eigenvectors
Exercises
1.
1.a
1.b
1.c
1.d
3.
5.
7.
7.a
7.b
7.c
7.d
7.d.i
7.d.ii
7.d.iii
9.
10.
10.a True/False.
10.c True/False.
21 Complex Eigenvalues
Exercises
1.
1.a
1.b
1.c
3.
5.
7.
7.a
7.b
8.
8.a True/False.
8.c True/False.
8.e True/False.
22 Properties of Determinants
Exercises
1.
3.
3.a
3.b
3.c
3.d
5.
7.
7.a
7.b
9.
12.
12.b
13.
13.a True/False.
13.c True/False.
13.e True/False.
13.g True/False.
V Orthogonality
23 The Dot Product in \(\R^n\)
Exercises
1.
1.a
1.b
1.c
1.d
1.e
3.
5.
7.
7.a
7.b
9.
11.
13.
15.
15.a True/False.
15.c True/False.
15.e True/False.
15.g True/False.
15.i True/False.
15.k True/False.
15.m True/False.
15.o True/False.
15.q True/False.
24 Orthogonal and Orthonormal Bases in \(\R^n\)
Exercises
1.
3.
3.a
3.b
3.c
3.d
5.
7.
7.a
7.b
7.c
9.
9.a True/False.
9.c True/False.
9.e True/False.
9.g True/False.
9.i True/False.
25 Projections onto Subspaces and the Gram-Schmidt Process in \(\R^n\)
Exercises
1.
1.a
1.b
3.
5.
7.
7.a
7.b
7.c
9.
9.a
9.b
11.
11.a
11.b
11.c
11.d
12.
12.a True/False.
12.c True/False.
12.e True/False.
12.g True/False.
26 Least Squares Approximations
Exercises
1.
1.a
1.b
3.
5.
5.a
5.b
5.c
5.d
6.
6.b
7.
7.a
7.b
7.c
7.d
7.e
9.
9.a True/False.
9.c True/False.
9.e True/False.
VI Applications of Orthogonality
27 Orthogonal Diagonalization
Exercises
1.
1.a
1.b
1.c
3.
5.
5.a
5.b
5.c
5.d
5.e
5.f
5.g
6.
7.
7.a
7.b
7.b.i
7.b.ii
7.b.iii
7.b.iv
7.b.v
7.c
8.
8.a True/False.
8.c True/False.
8.e True/False.
8.g True/False.
8.i True/False.
8.k True/False.
28 Quadratic Forms and the Principal Axis Theorem
Exercises
1.
1.a
1.b
1.c
3.
3.a
3.a.i
3.a.ii
3.a.iii
3.b
5.
5.a
5.b
5.c
5.d
7.
9.
11.
11.a
11.b
13.
14.
14.a True/False.
14.c True/False.
14.e True/False.
14.g True/False.
14.i True/False.
14.k True/False.
14.m True/False.
14.o True/False.
29 The Singular Value Decomposition
Exercises
1.
1.a
1.b
1.c
1.d
1.e
2.
2.a
3.
3.a
3.b
3.c
3.c.i
3.c.ii
3.c.iii
4.
4.b
5.
7.
9.
11.
11.a True/False.
11.c True/False.
11.e True/False.
11.g True/False.
30 Using the Singular Value Decomposition
Exercises
1.
1.a
1.b
1.c
1.d
3.
3.a
3.b
3.c
4.
4.b
4.b.iii
5.
5.a
5.b
5.c
5.d
5.e
6.
6.a
7.
7.a
7.b
7.c
9.
10.
10.a True/False.
10.c True/False.
10.e True/False.
VII Vector Spaces
31 Vector Spaces
Exercises
1.
1.a
1.b
1.c
4.
6.
8.
10.
12.
12.a
12.b
12.c
12.d
12.e
12.f
14.
14.a True/False.
14.c True/False.
14.e True/False.
14.g True/False.
14.i True/False.
32 Bases for Vector Spaces
Exercises
1.
1.a
1.b
1.c
1.d
3.
5.
7.
9.
10.
11.
13.
13.a True/False.
13.c True/False.
13.e True/False.
13.g True/False.
13.i True/False.
13.k True/False.
33 The Dimension of a Vector Space
Exercises
1.
3.
3.a
3.b
3.c
3.d
5.
7.
8.
8.b
9.
11.
11.a
11.b
12.
12.a True/False.
12.c True/False.
12.e True/False.
12.g True/False.
12.i True/False.
12.k True/False.
34 Coordinate Vectors and Coordinate Transformations
Exercises
1.
3.
5.
7.
7.a
7.b
7.b.i
7.b.ii
9.
9.a
9.b
9.c
11.
13.
13.a
13.b
15.
17.
17.a True/False.
17.c True/False.
17.e True/False.
17.g True/False.
17.i True/False.
35 Inner Product Spaces
Exercises
1.
3.
3.a
3.b
5.
5.a
5.b
7.
7.a
7.b
9.
9.a
9.b
11.
11.a
11.b
11.c
13.
13.a
13.b
13.c
15.
15.a
15.b
15.c
15.d
16.
17.
19.
19.a True/False.
19.c True/False.
19.e True/False.
19.g True/False.
19.i True/False.
19.k True/False.
19.m True/False.
19.o True/False.
36 The Gram-Schmidt Process in Inner Product Spaces
Exercises
1.
1.a
1.b
1.c
3.
3.a
3.b
5.
7.
7.a
7.b
9.
9.a True/False.
9.c True/False.
9.e True/False.
VIII Linear Transformations
37 Linear Transformations
Exercises
1.
3.
3.a
3.b
3.c
5.
5.a
5.b
7.
9.
10.
11.
13.
13.a
13.b
16.
16.a
16.b
19.
19.a True/False.
19.c True/False.
19.e True/False.
19.g True/False.
19.i True/False.
38 The Matrix of a Linear Transformation
Exercises
1.
1.a
1.b
1.c
1.d
1.e
1.f
1.g
3.
3.a
3.b
5.
7.
7.a
7.b
7.c
8.
8.b
8.d
9.
9.a True/False.
9.c True/False.
9.e True/False.
9.g True/False.
9.i True/False.
39 Eigenvalues of Linear Transformations
Exercises
1.
1.a
1.b
1.c
1.d
3.
3.a
3.b
3.c
5.
5.a
5.b
5.c
6.
6.a True/False.
6.c True/False.
6.e True/False.
6.g True/False.
40 The Jordan Canonical Form
Exercises
1.
1.a
1.b
1.c
1.d
3.
5.
7.
8.
8.e
9.
9.a
9.b
9.b.i
9.b.ii
9.b.iii
11.
11.a
11.b
13.
15.
17.
19.
21.
21.a
21.a.i
21.a.ii
21.b
23.
25.
25.a True/False.
25.c True/False.
25.e True/False.
25.g True/False.
25.i True/False.
25.k True/False.
25.m True/False.