Learn Linear Algebra - Free Course with Videos and Notes
Master Linear Algebra through structured video lessons, guided notes, example problems, and exam review resources. This free course covers systems of linear equations, matrices, vector spaces, eigenvalues and eigenvectors, and orthogonality, building understanding and confidence through clear, step-by-step instruction.
This series follows Linear Algebra and Its Applications by David C. Lay, Steven R. Lay, and Judi J. McDonald.
Linear Algebra Video Lessons with Guided Notes
Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.
Chapter 1 - Linear Systems of Equations
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In these videos, you'll learn what a system of linear equations is and how solutions can be interpreted geometrically. We'll introduce matrices, variables, and coefficients while developing a systematic approach for solving linear systems and determining whether solutions exist.
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In this video, you'll learn how elementary row operations can be used to solve systems of linear equations efficiently. We'll develop row echelon form and reduced row echelon form, identify pivot positions, and determine when a system has no solution, one solution, or infinitely many solutions.
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In these videos, you'll learn how systems of linear equations can be represented using vectors and vector equations. We'll explore linear combinations, geometric interpretations, and how vector equations provide a new perspective on solving linear systems.
Part 1
Part 2
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In this video, you'll learn how matrices provide a compact representation of linear systems. We'll develop the matrix equation Ax = b, connect it to vector equations, and investigate the conditions under which solutions exist. You'll also see how matrix inverses can be used to solve linear systems.
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In these videos, you'll learn how matrices are used to represent linear transformations. We'll construct standard matrices and investigate geometric transformations in ℝ², including reflections, projections, rotations, contractions, expansions, and shear transformations.
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In this video, you'll discover how linear algebra is used to model real-world systems in business, science, and engineering. We'll explore practical applications that demonstrate how matrices, vectors, and linear systems can be used to analyze data, solve problems, and make predictions.
Chapter 2 - Matrix Algebra
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In this video, you'll learn the fundamental operations used with matrices, including matrix addition, scalar multiplication, matrix multiplication, powers of matrices, and transposes. You'll also see how matrix multiplication can be used to represent linear transformations and solve a variety of linear algebra problems.
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Chapter 3 - Determinants
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In this video, you'll learn what determinants are and why they play an important role in linear algebra. We'll compute determinants of 2×2, 3×3, and larger matrices using cofactor expansion and the Rule of Sarrus, while exploring applications to matrix inverses, systems of equations, geometry, and eigenvalue problems.
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In this video, you'll learn the fundamental properties of determinants and how they can simplify determinant calculations. We'll investigate how elementary row operations affect determinants, use row reduction to compute determinants efficiently, develop important determinant formulas, and apply determinants to determine invertibility and linear independence.
Chapter 4 - Vector Spaces
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In this video, you'll learn about two of the most important subspaces associated with a matrix: the null space and the column space. We'll find null spaces by solving homogeneous systems, determine column spaces from matrix columns, and interpret how these spaces reveal key information about solutions to matrix equations.
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In this video, you'll learn how bases provide the building blocks for vector spaces. We'll define bases and standard bases, determine whether a set of vectors forms a basis, and develop methods for finding bases for both the null space and column space of a matrix.
Basis
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In this video, you'll learn how dimension measures the size and complexity of a vector space. We'll determine dimensions of common vector spaces and subspaces, connect dimension to the null space and column space of a matrix, and use the Rank Theorem to relate rank, nullity, and the number of matrix columns.
Chapter 5 - Eigenvalues and Eigenvectors
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In this video, you'll be introduced to one of the most important concepts in linear algebra: eigenvectors and eigenvalues. We'll explore their connection to linear transformations, learn how to determine whether a vector is an eigenvector or a number is an eigenvalue, and find bases for eigenspaces associated with a given eigenvalue.
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In this video, you'll learn how the characteristic equation allows us to find the eigenvalues of a matrix. We'll develop characteristic polynomials, solve for eigenvalues of both 2×2 and 3×3 matrices, examine important properties of eigenvalues, and investigate how row operations affect them.
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In this video, you'll learn how diagonalization transforms a matrix into a simpler form that reveals its underlying structure. We'll study the Diagonalization Theorem, determine when a matrix is diagonalizable, compute powers of matrices efficiently using diagonalization, and learn how to construct the matrices involved in a diagonalization.
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In this video, you'll see how eigenvalues and eigenvectors can be used to model real-world systems. We'll develop matrix models for population changes, introduce difference equations and transition matrices, and use dominant eigenvalues and eigenvectors to predict long-term growth rates and population distributions.
Chapters 6 - Orthogonality and Least Squares
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In this video, you'll learn how the inner product (dot product) provides a way to measure the relationship between vectors. We'll explore inner products, vector length and distance, unit vectors and normalization, and the concepts of orthogonal and orthonormal vectors that form the foundation for many later topics in linear algebra
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In this video, you'll extend the idea of orthogonal vectors to entire sets of vectors. We'll study orthogonal and orthonormal sets, learn how orthogonal bases simplify computations, express vectors as linear combinations of orthogonal basis vectors, compute orthogonal projections, decompose vectors into orthogonal components, and determine distances from vectors to lines through the origin.
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In this video, you'll generalize orthogonal projections from lines to higher-dimensional subspaces. We'll review projections onto lines, develop formulas for projecting vectors onto subspaces, find the closest point in a subspace to a given vector, decompose vectors into orthogonal components, and compute distances from vectors to subspaces.
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In this video, you'll learn the Gram-Schmidt process, a powerful algorithm for converting a set of linearly independent vectors into an orthogonal or orthonormal basis. We'll develop the Gram-Schmidt algorithm, construct orthogonal and orthonormal bases for subspaces, find orthogonal bases for column spaces of matrices, and introduce QR factorization and its many practical applications.
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In this video, you'll learn how to find the best approximate solution to systems that have no exact solution. We'll develop the least-squares method, derive and solve the normal equations, compute least-squares error, construct lines of best fit for data, and use QR factorization to efficiently solve least-squares problems.
Chapter 7 - Applications
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In this video, you'll learn one of the most powerful matrix factorizations in linear algebra. We'll develop the singular value decomposition A = UΣV^T, learn how singular values are related to the eigenvalues of A^T A, construct the matrices U, Σ, and V, identify left and right singular vectors, and interpret SVD geometrically as a combination of rotations, reflections, and scaling transformations.
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In this video, you'll learn how linear algebra can be used to analyze and simplify data. We'll motivate the need for PCA, organize data into matrices, center data around their means, construct covariance matrices, identify principal directions using eigenvalues and eigenvectors, determine how much variation each principal component explains, and compute principal component scores to reveal the most important patterns in a dataset.
Copyright & Usage
© Understand The Math LLC. All Rights Reserved. These materials are freely provided for non-commercial educational use. Redistribution is permitted with proper attribution to UnderstandTheMath.com.
Linear Algebra Example Problems
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Linear Algebra Exam Review
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Linear Algebra Exam Review Page
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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.