To use linear algebra in data science, learn how vectors and matrices represent data and models, then focus on least squares, orthogonality, eigenvalues, singular value decomposition (SVD), and low-rank approximation. These ideas connect directly to regression, principal component analysis (PCA), and other methods that find structure in data. Start with an introductory text if you are new to the subject; move on to an applied course once you have the basics.
Why linear algebra matters in data science
A data table can be represented as a matrix: rows often represent observations, while columns represent features. Vectors can represent an individual observation, a model’s parameters, or a direction in the data. Matrix operations then express transformations and systems of equations compactly.
This is more than notation. Matrix methods help describe how models fit data, how dimensions can be reduced, and how algorithms can be organized for computation. MIT’s 18.065 course connects matrix methods with machine learning, probability, statistics, and optimization.
Which linear algebra concepts should you learn?
Vectors, matrices, and matrix multiplication
Learn how to represent observations and collections of observations, and how multiplication by a matrix changes or combines them. This gives you the language to write many data problems as compact equations rather than separate calculations.
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Least squares and projections
Real data often do not satisfy a model equation exactly. Least squares finds a model whose predictions are as close as possible to the observations under a squared-error criterion. It is a practical bridge from linear algebra to regression. Projections explain geometrically how a fitted result relates to the space spanned by a model’s features.
Subspaces and orthogonality
A subspace captures the directions available to a model or data representation. Orthogonality describes perpendicular directions and makes projections and decompositions easier to reason about. These ideas help explain why some representations separate components cleanly, though not every data science workflow requires advanced spectral methods.
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Eigenvalues and eigenvectors
For a square matrix, an eigenvector is a direction that the matrix maps to itself apart from a scale change; its eigenvalue gives that scale. Eigenvectors help reveal important directions in certain transformations and methods. They are useful to understand, but they are not a prerequisite for every modeling task.
SVD, PCA, and low-rank approximation
Singular value decomposition (SVD) expresses a matrix using singular vectors and singular values. It provides a route to principal component analysis and to low-rank approximations: simpler matrices that retain prominent structure while using fewer dimensions. MIT’s 18.065 reading sequence places SVD alongside principal components and best rank-k approximation, making these natural topics to study together.
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Numerical methods and computational cost
Data matrices can be large, so understanding a mathematical method is only part of the work. Numerical linear algebra addresses how computations behave in practice, while approaches such as randomized matrix multiplication can help with large-scale calculations. MIT includes both numerical methods and randomized matrix multiplication in its applied course topics.
A practical learning sequence
- Build the basics. Learn vectors, matrices, matrix multiplication, and how to express simple systems of equations.
- Connect equations to fitting. Study least squares, projections, and the geometry of subspaces; relate them to a small regression problem.
- Study structure and decomposition. Learn orthogonality and eigenvalues, then move to SVD, PCA, and low-rank approximation.
- Add computational perspective. Explore norms and numerical methods, and consider why the cost of working with a large matrix matters.
- Practice with data tasks. Write a small regression problem as a matrix equation, compare a data matrix with a low-rank approximation, and use an SVD or PCA example to see how a high-dimensional table can be summarized.
These practice activities follow from the topics and applications covered by the cited courses and texts; they are suggested exercises, not reported independent test results.
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Choose a resource that matches your background
| Resource | Best fit | Background and content |
|---|---|---|
| Stanford’s Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares | Beginners and self-studiers | Intended for readers with little prior linear algebra; covers vectors, matrices, least squares, and applications including data fitting and machine learning. |
| MIT OpenCourseWare 18.065, Matrix Methods in Data Analysis, Signal Processing, and Machine Learning | Learners who already know linear algebra | Lists 18.06 Linear Algebra as a prerequisite. The Spring 2018 course includes lecture videos, problem sets, labs, and a project, with applications to machine learning, probability, statistics, and optimization. |
| Gilbert Strang’s Linear Algebra and Learning from Data | Learners who want a structured textbook alongside an applied course | Identified as the 18.065 textbook. Course readings cover SVD, principal components, least squares, norms, and numerical methods. |
The resource descriptions establish their intended audiences and subject matter, not comparative learning outcomes or completion rates. MIT’s syllabus is from Spring 2018, so check the course page for the materials currently available. MIT also lists related linear algebra textbooks, including Introduction to Linear Algebra and Linear Algebra for Everyone.
What linear algebra can—and cannot—promise
The cited course and textbook materials explain how linear algebra connects to data analysis and machine learning, but they do not quantify how much studying it improves data science performance, employability, or course completion. Treat it as foundational mathematical knowledge that helps you understand and work with methods—not as a guaranteed outcome or a substitute for statistics, programming, and domain knowledge.
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