Learn Linear Algebra
with Clear and Detailed
Example Problems!
Welcome to our Linear Algebra Example Problems page! Here, you'll find a collection of solved problems designed to deepen your understanding of fundamental concepts in linear algebra. Each problem is broken down into clear, step-by-step solutions to help you master key topics and build confidence in solving similar problems on your own.
Explore Example Problems:
Solving systems of linear equations
Matrix operations and determinants
Eigenvalues and eigenvectors
Orthogonal projections and vector spaces
The Gram-Schmidt process and least-squares problems
Video Tutorials:
Each example links to a detailed video tutorial where we walk through the solution and key concepts. Perfect for preparing for exams, brushing up on skills, or tackling tricky homework questions.
Start exploring the examples below and click on the videos to watch step-by-step tutorials. Let’s make mastering linear algebra easier than ever!
Other Linear Algebra Resources: Linear Algebra Full Course and Linear Algebra Exam Review
Videos
1: Linear Systems of Equations
Find Solutions To A Linear System: No Solution, One Solution, Or Infinitely Many Solutions
How To Identify Echelon Form And Reduced Echelon Forms In Matrices
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
How to Find the Geometric Description of a Span
How to Determine if a Vector is in the Span of Other Vectors
How to Write Matrix and Vector Equations for Linear Systems
How to Determine When Ax=b Has a Solution
How to Solve Ax=0 and Ax=b Using Augmented Matrices
How to Find Values of H for Linearly Dependent Vectors
How to Determine Linear Independence or Dependence by Inspection
2. Linear Transformations
How to Find the Image of a Vector with Standard Matrix
How to Find a Vector When Its Image is Known
How to Determine if a Vector is in the Range of a Transformation
How to Find the Domain and Codomain of a Linear Transformation
How to Determine if T(x) is a Linear Transformation
How to Find the Standard Matrix of a Linear Transformation
How to Find the Image of a Vector Under a Linear Transformation
How to Compute T(x) Using a Linear Combination of Basis Vectors
How to Find the Standard Matrix for a Composite Reflection Transformation
How to Find the Standard Matrix for Rotation and Stretching
How to Determine if a Transformation is One-to-One and/or Onto
3. Matrix Algebra
How to Multiply and Transpose Matrices: Step-by-Step Example
How to solve a Linear System Using the Inverse of a Matrix
How to Find the Inverse of a 3x3 Matrix
How to Simplify Matrix Expressions
How to Find the Inverse of a Linear Transformation
4. Determinants
How to Compute a Determinant Using Cofactor Expansion
How to Compute a Determinant Using Row Reduction
How to Compute Determinants Using Matrix Properties
How to Compute Determinants for Matrix Products and Powers
How to Determine if a Matrix is Invertible Using Determinants
5. Vector Spaces
How to Determine if a Set is a Subspace of R^2
How to Determine if a Vector is in the Null Space and/or Column Space of a Matrix
How to Find the Null Space and Column Space of a Matrix | Basis and Geometric Description
How to Determine if a Set of Vectors is a Basis for R^3
How to Find the Dimension of a Subspace Spanned by a Set of Vectors
How to Find the Rank and Nullity of a Matrix
How to Use the Rank-Nullity Theorem to Find the Rank of a Matrix
6. Eigenvalues and Eigenvectors
How to Determine if a Vector is an Eigenvector and Find it's Eigenvalue
How to Find the Eigenvalues of a 2x2 Matrix
How to Find Eigenvalues and Eigenvectors of a 3x3 Matrix
How to Find the Eigenvalues of a Matrix’s Transpose, Inverse, and Power
How to Determine if a Matrix is Diagonalizable
How to Compute Matrix Powers Using Diagonalization
7. Orthogonality
How to Find a Unit Vector in the Direction of a Given Vector
How to Find the Distance Between Two Vectors
How to Determine if a Set of Vectors is Orthogonal and Orthonormal
How to Express a Vector as a Linear Combination of Orthogonal Vectors
How to Find the Orthogonal Projection of a Vector onto Another Vector
How to Find the Orthogonal Projection of a Vector Onto a Subspace
How to Find an Orthogonal Basis and QR Factorization of a Matrix
How to Find the Least Squares Line that Best Fits a Set of Data Points