Learn Linear Algebra Through Example Problems
Master systems of linear equations, matrix algebra, determinants, vector spaces, eigenvalues and eigenvectors, and orthogonality through step-by-step example problems that build understanding, strengthen problem-solving skills, and prepare you for homework, quizzes, and exams.
Chapter 1: Linear Systems of Equations Example Problems
Practice solving linear systems, working with matrix equations, understanding linear independence, and building the foundation for linear transformations.
Solving Systems
Find Solutions To A Linear System: No Solution, One Solution, Or Infinitely Many Solutions
How to Solve Linear Systems: No Solution, Unique Solution, or Infinite Solutions Explained
How to Solve Linear Systems Using Gauss-Jordan Elimination
How to Find the General Solution of a Linear System Using an Augmented Matrix
Understanding Structure
How To Identify Echelon Form And Reduced Echelon Forms In Matrices
How to Write Matrix and Vector Equations for Linear Systems
How to Determine When Ax=b Has a Solution
How to Find the Geometric Description of a Span
How to Determine if a Vector is in the Span of Other Vectors
How to Find Values of H for Linearly Dependent Vectors
How to Determine Linear Independence or Dependence by Inspection
Computing with Transformations
How to Find the Standard Matrix of a Linear Transformation
How to Find the Image of a Vector with Standard Matrix
How to Find the Image of a Vector Under a Linear Transformation
How to Find a Vector When Its Image is Known
How to Compute T(x) Using a Linear Combination of Basis Vectors
How to Find the Standard Matrix for a Composite Reflection Transformation
Chapter 2: Matrix Algebra Example Problems
Practice matrix operations, inverses, matrix equations, and linear transformations while developing the matrix techniques used throughout linear algebra.
Chapter 3: Determinant Example Problems
Practice computing determinants using cofactor expansion, row reduction, and matrix properties while developing techniques used to analyze matrices and invertibility.
Chapter 4: Vector Spaces Example Problems
Practice working with subspaces, null spaces, column spaces, bases, dimension, rank, and nullity while developing a deeper understanding of the structure of vector spaces.
Chapter 5: Eigenvalues and Eigenvector Example Problems
Practice finding eigenvalues and eigenvectors, determining whether a matrix is diagonalizable, and using diagonalization to simplify matrix computations.
Chapter 6: Orthogonality Example Problems
Practice working with orthogonal and orthonormal vectors, projections, orthogonal bases, QR factorization, and least-squares methods while developing powerful tools for approximation and data analysis.
Chapter 7: Application Example Problems
Explore powerful applications of linear algebra through singular value decomposition (SVD) and principal component analysis (PCA), two techniques widely used in data analysis, machine learning, and scientific computing.
About Dr. Cheryl Hile
Understand The Math was created by Dr. Cheryl Hile, who earned a Ph.D. in Engineering Science and Applied Mathematics from Northwestern University. She has more than 25 years of experience teaching mathematics at the university level, including 20 years at Penn State. The lessons and guided notes on this site are based on materials she developed for her university courses.