Linear Algebra
I’m new to linear algebra
Start with MIT’s independent-study course and follow its lectures, notes, and problems in sequence.
Start with: Linear Algebra
RedLectern topic
Linear Algebra
Core ideas and methods for systems of equations, vector spaces, linear transformations, and matrix analysis.
For: Learners at different starting points, from those meeting vectors and matrices for the first time to those revisiting the subject for further study.
After this page you can: Choose a useful learning path and explain how a linear algebra method represents or solves a problem.
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Part of: Mathematics
About Linear Algebra
Linear algebra studies systems of linear equations and the properties of matrices. It also develops ideas such as vector spaces, bases, and eigenvalues, which help describe how matrices act on vectors. You can place the subject within mathematics and follow related paths to statistics, calculus, or geometry.
A useful way to study is to work examples, then revisit what the calculations mean: begin with systems and elimination, build the language of spaces and bases, and continue to transformations, projections, and eigenvalues. Choose a route based on your current goal; you do not need to master every topic before applying the ideas.
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Resource Index: Linear Algebra
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Use the resource index to find a focused topic, then try its problems and check your work.
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Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
I want to connect it to data
After reviewing core linear algebra, explore MIT’s course on matrix methods and their applications to data analysis.
Start with: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
Learn the foundations
1.Solve systems with matricesStart with Ax = b, elimination, and the role of the column space in determining when a solution exists.
Start with Ax = b, elimination, and the role of the column space in determining when a solution exists.
- Column Space and Nullspace
The lesson explains when Ax = b has a solution and which values of x solve Ax = 0.
Nullspace
The set of inputs x that solve Ax = 0.
Try it
In your own words, state the condition on b that makes Ax = b solvable.
Ax = b is solvable exactly when b belongs to the column space of A.
2.Build the language of vector spacesStudy subspaces, independence, bases, and dimension to describe the structure behind collections of vectors.
Study subspaces, independence, bases, and dimension to describe the structure behind collections of vectors.
- Independence, Basis and Dimension
The lesson defines basis and dimension.
Basis
A set of vectors, as few as possible, whose combinations produce all vectors in the space.
Dimension
The number of vectors in a basis for a space.
Try it
Explain what a basis provides for a vector space.
A basis is a set of vectors whose combinations produce all vectors in the space, with no more vectors than needed.
3.Understand transformations and determinantsConnect matrix multiplication to linear transformations, and use determinants to characterize important properties of square matrices.
Connect matrix multiplication to linear transformations, and use determinants to characterize important properties of square matrices.
- Linear Transformations and their Matrices
The lesson asks what the linear transformation represented by matrix multiplication does to vectors in a space.
Determinant
A value useful for determining whether a matrix is singular and computing its inverse.
Try it
Describe what a matrix does when it multiplies an input vector.
It produces an output vector, and the resulting rule can be studied as a linear transformation.
4.Use orthogonality and least squaresUse projections to connect orthogonality with least-squares fitting and approximation.
Use projections to connect orthogonality with least-squares fitting and approximation.
- Projection Matrices and Least Squares
The lesson presents least squares as finding the projection of a vector.
Orthogonal projection
The projection of a vector onto a subspace along a direction perpendicular to that subspace.
Least squares
A method for fitting a line or curve to data that can be viewed as finding a vector’s projection.
Try it
According to the projection view of least squares, what geometric operation does the method use?
It finds the projection of a vector.
5.Study eigenvalues and eigenvectorsExplore how eigenvalues and eigenvectors describe the effect of multiplying a vector by a matrix.
Explore how eigenvalues and eigenvectors describe the effect of multiplying a vector by a matrix.
- Eigenvalues and Eigenvectors
The lesson introduces eigenvalues and eigenvectors and offers problems with solutions.
Eigenvector
A nonzero vector whose product with a matrix points in the same direction as the vector.
Try it
In the relation Ax = λx, what does it mean for x to be an eigenvector?
It means Ax points in the same direction as x; λ is the corresponding eigenvalue.
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Books
- Introduction to Linear Algebra
Introduction to Linear Algebra
Gilbert Strang · 2016
A main-route text for learners who want geometric intuition alongside matrix techniques and abstract ideas.
- Linear Algebra Done Right
Linear Algebra Done Right
Sheldon Axler · 2024
A focused follow-on for readers ready to study the structure of linear operators and proof-based linear algebra.
- Linear Algebra and Its Applications
Linear Algebra and Its Applications
David C. Lay, Steven R. Lay, and Judi J. McDonald · 2016
A useful applied companion for learners who want the standard course topics organized around computational examples.
- Numerical Linear Algebra
Numerical Linear Algebra
Lloyd N. Trefethen and David Bau III · 1997
A specialist continuation for understanding how linear algebra behaves in computation.
- Linear Algebra
Linear Algebra
Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence · 2019
A theoretical course text for readers who want a more formal treatment of the subject.
Every free Linear Algebra course and book on RedLectern
Free Linear Algebra online courses
- A Vision of Linear AlgebraMIT RES.18-010 · Browse materials · Video
- Algebra IMIT 18.701 · Browse materials
- Algebra IIMIT 18.702 · Browse materials
- Calculus Revisited: Complex Variables, Differential Equations, and Linear AlgebraMIT RES.18-008 · Browse materials · Video
- Differential EquationsMIT 18.03 · Browse materials · Video
- Differential EquationsMIT 18.03SC · Browse materials · Video
- Differential EquationsMIT ES.1803 · Browse materials
- Engineering Math: Differential Equations and Linear AlgebraMIT 2.087 · Browse materials · Video
Show all 22 coursesShow fewer courses
- Honors Differential EquationsMIT 18.034 · Browse materials
- Introduction to Functional AnalysisMIT 18.102 · Browse materials
- Introduction to Functional AnalysisMIT 18.102 · Browse materials · Video
- Introduction to Representation TheoryMIT 18.712 · Browse materials
- Linear AlgebraMIT 18.06 · Browse materials · Video
- Linear AlgebraMIT 18.06SC · Browse materials · Video
- Linear AlgebraMIT 18.700 · Browse materials
- Linear Algebra 1 (English)TU Delft OpenCourseWare · Browse materials · Video
- Seminar in Algebra and Number Theory: Rational Points on Elliptic CurvesMIT 18.704 · Browse materials
- Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of VariationsMIT 2.035 · Browse materials
- Topics in Geometry: Dirac GeometryMIT 18.969 · Browse materials
- Honors Differential EquationsMIT 18.034 · Open on OCW
- Linear AlgebraMIT 18.06SC+summer_2019 · Open on OCW
- Linear Algebra - Communications IntensiveMIT 18.06CI · Open on OCW
How these courses are chosen
Listed as Linear Algebra courses:
- those whose title contains “Linear Algebra”, or
- those tagged “Linear Algebra” as one of at most 2 topic tags, or as the first of 3.
Free Linear Algebra textbooks and books
- Sheldon Axler · Read at linear.axler.net ↗ · Cited by 2 MIT courses
- Robert A. Beezer · Read at linear.pugetsound.edu ↗
- Joel G. Broida · Read at cseweb.ucsd.edu ↗
- Marilena Beckstrand · Read at math.libretexts.org ↗
- John W. Sylvester · Read at sites.ualberta.ca ↗
- Keith R. Matthews · Read at www.numbertheory.org ↗
- Edwin H. Connell · Read at www.math.miami.edu ↗
- Dan Margalit · Read at sites.math.duke.edu ↗
Show all 18 booksShow fewer books
- Lieven Vandenberghe Stephen Boyd · Read at web.stanford.edu ↗
- Jim Hefferon · Read at hefferon.net ↗
- Sergei Treil · Read at sites.google.com ↗
- Sergei Winitzki · Read at sites.google.com ↗
- W. Keith Nicholson · Read at open.umn.edu ↗
- Kenneth Kuttler · Read at open.umn.edu ↗
- Isaiah Lankham · Read at www.math.ucdavis.edu ↗
- Peter Selinger · Read at www.mathstat.dal.ca ↗
- Leon Q. Brin · Read at open.umn.edu ↗
- David Austin · Read at scholarworks.gvsu.edu ↗