Introduction to Linear Algebra

list of linear algebra problems

Introduction to Linear Algebra

Some problems and solutions by the topics that are taught in the undergraduate linear algebra course (Math 2568) in the Ohio State University.

The number of chapters/sections are based on the textbook Introduction to Linear Algebra, 5th edition,
by L.W. Johnson, R.D. Riess, and J.T. Arnold.


The problems with ★ are more advanced problems than typical exercise problems, but not necessarily unaccessible.

Chapter 1. Matrices and Systems of Linear Equations

1.1 Introduction to Matrices and Systems of linear equations

1.2 Echelon Form and Gaussian-Jordan Elimination

1.3 Consistent Systems of linear Equations

1.5 Matrix Operations

1.6 Algebraic Properties of Matrix operations

1.7 Linear Independence and Nonsingular Matrices

1.9 Matrix Inverses and Their Properties

Midterm Exam 1 (covers Chapter 1)

Chapter 2. Vectors in 2-Space and 3-Space

2.1 Vectors in The Plane

2.2 Vectors in Space

2.3 The Dot Product and The Cross Product

Chapter 3. The Vector Space $\R^n$

3.2 Vector Space Properties of R^n

3.3 Examples of Subspaces

3.4 Bases for Subspaces

3.5 Dimension

3.6 Orthogonal Bases for Subspaces

3.7 Linear Transformation from $R^n to R^m$

Chapter 4. The Eigenvalue Problem

4.1 The Eigenvalue Problem for 2×2 Matrices

4.2 Determinants and the Eigenvalue Problem

4.3 Elementary Operations and Determinants

(This section is not covered in Math 2568.)

4.4 Eigenvalues and Characteristic Polynomial

4.5 Eigenvectors and Eigenspaces

4.6 Complex Eigenvalues and Eigenvectors

4.7 Similarity Transformations and Diagonalization

Extra Problems on Eigenvalues and Diagonalization

Chapter 5. Vector Spaces and Linear Transformations

5.2 Vector Spaces

5.3 Subspaces

5.4 Linear Independence, Bases, and Coordinates

The following sections are not covered in Math 2568 at OSU.

5.6 Inner-Product Spaces, Orthogonal Bases, and Projections

5.7 Linear Transformations

5.8 Operations with Linear Transformations

5.10 Change of Basis and Diagonalization.

Chapter 6. Determinants

6.4 Cramer’s Rule

6.5 Applications of Determinants: Inverses and Wronksians