Linear algebra learning recommendations

This page provides subject-specific learning recommendations for the subject linear algebra. See all our learning recommendations pages |

Linear algebra is an important mathematical topic with wide applicability to the natural and social sciences, and is the only topic outside of the calculus sequences that is widely studied by undergraduates who do not intend to major in mathematics. Although some computational background related to linear algebra is covered in school syllabi, the concepts of linear algebra at the college level can still be very difficult for students, because of the focus of most curricula on preparing students for calculus to the exclusion of other parts of mathematics. The course may be taken as follows:

• Undergraduates typically take linear algebra after completing single variable calculus and multivariable calculus, although calculus is not formally a prerequisite for linear algebra. In some places, linear algebra and multivariable calculus are taught together as a third semester calculus course. In some places, they are separate semester-length or quarter-length courses.
• Some undergraduates who intend to major in mathematics may not take a separate linear algebra course but may study parts of it in their analysis and abstract algebra courses.
• Some students who finish single variable calculus in 11th grade (junior year of high school) may study linear algebra in 12th grade (senior year of high school). See our high school mathematics page for descriptions of common tracks taken by students doing mathematics in high school.
• Other students might do a few linear algebra topics (solving systems of equations, matrices, determinants) as part of algebra courses in high school.

This page includes some guidance on who needs to learn linear algebra, what parts of linear algebra are important,resource suggestions for people interested in learning linear algebra.

See also:

• Calculus learning recommendations
• Multivariable calculus learning recommendations
• Online mathematics learning resources
• Mathematics reading recommendations
• Learn mathematics well

Parts of linear algebra and their relative importance

Part of linear algebra	Details	Who needs to learn this?
Foundation	Solving systems of linear equations (using Gauss-Jordan elimination via row reduction), basics of vectors and vector spaces, linear combinations, linear relations, spans, spanning sets, subspaces, matrices as encoding linear transformations, geometric interpretation of linear transformations, matrix multiplication, and inversion.	Anybody who needs to learn algebra for any purpose.
Regression	Ordinary least squares regression and its matrix implementation.	People learning linear algebra for the purpose of applying it in any context, such as the sciences (natural and social) or any context that involves experimentation, measurement, and statistics.
Advanced computational topics	Determinants, eigenvalues, QR factorization, Gram-Schmidt process, Gram-Schmidt factorization, singular value decomposition, etc.	Some application contexts, such as machine learning, and some sciences that rely on heavy statistical machinery.
Theoretical linear algebra	Same as the foundation topics, but done right from the beginning in the context of abstract spaces and with generic proofs rather than relying on the language of matrices.	Useful if pursuing higher mathematics.

Recommendations

Recommendations for learning theoretically oriented linear algebra

• Linear Algebra Done Right by Sheldon Axler (Amazon)
• Center of Math linear algebra text

Recommendations for learning linear algebra with an application orientation (includes general purpose recommendations)

• The foundation topics and regression are covered in Vipul Naik's online Math 196 notes. These provide considerable insight into linear algebra from a statistical application-oriented view, without including any statistics. However, due to the paucity of worked examples, these are best studied in conjunction with a book that has worked examples.
• Center of Math linear algebra text
• Linear Algebra with Applications (5th Edition) (Amazon) by Otto Bretscher is a reasonably decent book (it is also synced with Vipul Naik's online notes, though it may not be worth buying just for that purpose).