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 Solve Matrix Equations

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

How to Diagonalize a Matrix

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