An Inquiry-Based Introduction to Linear Algebra and Applications

Appendix B Answers and Hints for Selected Exercises

I Systems of Linear Equations
1 Introduction to Systems of Linear Equations
Exercises

2 The Matrix Representation of a Linear System
Exercises

3 Row Echelon Forms
Exercises

4 Vector Representation
Exercises

5 The Matrix-Vector Form of a Linear System
Exercises

6 Linear Dependence and Independence
Exercises

7 Matrix Transformations
Exercises

II Matrices
8 Matrix Operations
Exercises

9 Introduction to Eigenvalues and Eigenvectors
Exercises

10 The Inverse of a Matrix
Exercises

11 The Invertible Matrix Theorem
Exercises

III The Vector Space ${\mathbb{R}}^n$
12 The Structure of ${\mathbb{R}}^n$
Exercises

13 The Null Space and Column Space of a Matrix
Exercises

14 Eigenspaces of a Matrix
Exercises

15 Bases and Dimension
Exercises

16 Coordinate Vectors and Change of Basis
Exercises

IV Eigenvalues and Eigenvectors
17 The Determinant
Exercises

18 The Characteristic Equation
Exercises

19 Diagonalization
Exercises

20 Approximating Eigenvalues and Eigenvectors
Exercises

21 Complex Eigenvalues
Exercises

22 Properties of Determinants
Exercises

V Orthogonality
23 The Dot Product in ${\mathbb{R}}^n$
Exercises

24 Orthogonal and Orthonormal Bases in ${\mathbb{R}}^n$
Exercises

25 Projections onto Subspaces and the Gram-Schmidt Process in ${\mathbb{R}}^n$
Exercises

26 Least Squares Approximations
Exercises

VI Applications of Orthogonality
27 Orthogonal Diagonalization
Exercises

28 Quadratic Forms and the Principal Axis Theorem
Exercises

29 The Singular Value Decomposition
Exercises

30 Using the Singular Value Decomposition
Exercises

VII Vector Spaces
31 Vector Spaces
Exercises

32 Bases for Vector Spaces
Exercises

33 The Dimension of a Vector Space
Exercises

34 Coordinate Vectors and Coordinate Transformations
Exercises

35 Inner Product Spaces
Exercises

36 The Gram-Schmidt Process in Inner Product Spaces
Exercises

VIII Linear Transformations
37 Linear Transformations
Exercises

38 The Matrix of a Linear Transformation
Exercises

39 Eigenvalues of Linear Transformations
Exercises

40 The Jordan Canonical Form
Exercises