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Linear Algebra and Its Applications
Renowned professor and author Gilbert Strang demonstrates that linear algebra is a fascinating subject by showing both its beauty and value. While the mathematics is there, the effort is not all concentrated on proofs. Strang's emphasis is on understanding. He explains concepts, rather than deduces. This book is written in an informal and personal style and teaches real mathematics. The gears change in Chapter 2 as students reach the introduction of vector spaces. Throughout the book, the theory is motivated and reinforced by genuine applications, allowing pure mathematicians to teach applied mathematics.
487 pages, Hardcover
First published April 1, 1976
About the author
Gilbert Strang
38 books201 followersWilliam Gilbert Strang (born November 27, 1934), usually known as simply Gilbert Strang or Gil Strang, is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. He has made many contributions to mathematics education, including publishing seven mathematics textbooks and one monograph. Strang is the MathWorks Professor of Mathematics at the Massachusetts Institute of Technology. He teaches Introduction to Linear Algebra and Computational Science and Engineering and his lectures are freely available through MIT OpenCourseWare.
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Displaying 1 - 30 of 60 reviews
November 23, 2012
I wanted to hate this text, because Strang has one of those obnoxious voices where he pretends to teach you math like you're just two bros shootin the shit at the corner store:
Fortunately, Strang overcomes his cutesy voice by having a really very solid presentation of the material: much better than that of my professor at an unnamed University currently charging extortionist fees for the privilege of learning math from a professor very good at research who hasn't actually spoken to another human being outside of peer-reviewed journals since that summer of 1961 when a girl mistook him for someone else at a coffee shop.
I also would complain, briefly, about Strang's notation, which struck me as about the same level as his authorial voice; which is to say, somewhat limited in rigor for the sake of plainness. I guess this is better than the opposite problem but, really, who uses a superscript H for the conjugate transpose of a matrix? An asterisk (*) is used in every single other text I've come across.
But complaints aside, Strang is to be commended for focusing on the conceptual framework of linear algebra, with matrices as general linear operators, and eigenvalues/eigenvectors of matrix as fundamental properties of systems' dynamical evolution, instead of just throwing all the math at us and hoping that we can sort it out.
Hey dude, I know this is a l'il bit crazy, but to study general similarity transform matrices M, let's first check out some totally restricted examples where M has gotta be unitary. Weird, right? But I'll bet once you see Schur's Lemma you'll see why this was so rad. [not an actual quote]Lots of contemporary texts try to pull this off, and it's just too Mr.-Rogers-y for my taste.
Fortunately, Strang overcomes his cutesy voice by having a really very solid presentation of the material: much better than that of my professor at an unnamed University currently charging extortionist fees for the privilege of learning math from a professor very good at research who hasn't actually spoken to another human being outside of peer-reviewed journals since that summer of 1961 when a girl mistook him for someone else at a coffee shop.
I also would complain, briefly, about Strang's notation, which struck me as about the same level as his authorial voice; which is to say, somewhat limited in rigor for the sake of plainness. I guess this is better than the opposite problem but, really, who uses a superscript H for the conjugate transpose of a matrix? An asterisk (*) is used in every single other text I've come across.
But complaints aside, Strang is to be commended for focusing on the conceptual framework of linear algebra, with matrices as general linear operators, and eigenvalues/eigenvectors of matrix as fundamental properties of systems' dynamical evolution, instead of just throwing all the math at us and hoping that we can sort it out.
December 19, 2008
Gilbert Strang has an uncontested position among my top 4 fantasy grandfathers. This man exudes a love for math and you can't help but be infected when you read his books.
Never does Strang slam an equation on the table and immediately expect you to calculate. Nay! Each equation is given its proper introduction. Like any proper courtship, the reader is led to learn about the equation's backgrounds, goals, and common activities before any intermingling takes place. Each exercise thereafter is a tiny little journey at the end of which something new is discovered.
Linear Algebra is a crucial topic for understanding probability, quantum mechanics, and a whole slew of other topics in engineering and mathematics. If you're interested in any of these areas, preparation with this book would not only be helpful but quite fun along the way.
This is mathematics education done right.
Never does Strang slam an equation on the table and immediately expect you to calculate. Nay! Each equation is given its proper introduction. Like any proper courtship, the reader is led to learn about the equation's backgrounds, goals, and common activities before any intermingling takes place. Each exercise thereafter is a tiny little journey at the end of which something new is discovered.
Linear Algebra is a crucial topic for understanding probability, quantum mechanics, and a whole slew of other topics in engineering and mathematics. If you're interested in any of these areas, preparation with this book would not only be helpful but quite fun along the way.
This is mathematics education done right.
June 8, 2013
also check the author's class video on MIT opencourseware
August 14, 2013
A very confusing text book and an unpleasant read. I have found some textbooks quite good, but this one is not.
September 4, 2011
I hated and neglected linear algebra, then spent the rest of college wishing I'd learned it the first time around.
December 18, 2013
Gilbert Strang begins his classic text with a brief and enthusiastic explanation of the nature of linear algebra. He admires the natural beauty of this mathematical language. The student is easily convinced of these claims. Strang writes in a very colloquial, natural way - though his textbook does not flow as easily as his lectures. Working his problem sets, one practices both theoretical and computational algebraic skills. A very lovely, graceful, poetic examination of linear algebra from a very applied background.
February 20, 2017
Introduction
Gilbert Strang’s Linear Algebra and Its Applications is a classic undergraduate textbook that presents linear algebra as a practical, intuitive and intellectually rewarding subject. Rather than beginning with abstract definitions and lengthy formal proofs, Strang develops the subject through concrete matrices, systems of equations and geometric interpretations. His approach reflects a central conviction: linear algebra becomes much easier to understand when students can see what mathematical operations mean, rather than simply memorising the rules for carrying them out.
The book is particularly valuable for students of mathematics, engineering, physics, computer science and other quantitative disciplines. It connects fundamental mathematical concepts with applications in scientific computing, network analysis and engineering, making it clear that linear algebra is not merely a collection of computational techniques but a framework for understanding relationships, transformations and complex systems.
Strang’s approachable writing style, emphasis on conceptual understanding and focus on practical applications have helped establish his textbooks as influential resources for generations of students. Nevertheless, the book’s accessible approach also raises questions about the balance between intuition, computation and formal mathematical rigour.
Content and Organisation
A defining feature of Strang’s approach is that he begins with concrete problems. Gaussian elimination, the solution of systems of linear equations and the structure of matrices provide an accessible entry point into the subject. These topics allow students to develop computational confidence before encountering more abstract ideas.
From this foundation, the book explores linear combinations, matrix multiplication, inverses, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and related concepts. The progression demonstrates how apparently separate techniques are connected through a common mathematical structure.
This organisation is particularly effective because it places computation and interpretation alongside one another. Students learn how to solve a system of equations while also considering when a solution exists, whether it is unique, and how the underlying equations determine the possible outcomes. Such questions encourage readers to look beyond numerical answers and understand the structure of the problems they are solving.
The book’s title is also significant. Its emphasis on applications distinguishes it from texts that concentrate primarily on abstract algebraic structures. Strang repeatedly demonstrates how linear algebra can be used to represent and analyse problems arising in science and engineering, giving the material a clear purpose beyond examination preparation.
The Four Fundamental Subspaces: The Book’s Central Strength
One of Strang’s most influential contributions to undergraduate linear algebra education is his explanation of the four fundamental subspaces of a matrix: the column space, row space, nullspace and left nullspace.
These spaces are often introduced as separate definitions, but Strang presents them as interconnected parts of a unified mathematical picture. The column space describes the possible outputs of a matrix transformation, while the nullspace identifies the inputs that the matrix maps to zero. The row space and left nullspace provide complementary information about the equations themselves and the relationships among their constraints.
This viewpoint helps students understand why systems of linear equations may have no solution, one solution or infinitely many solutions. It also clarifies the relationships between rank, linear independence and the dimensions of the relevant spaces.
The discussion is particularly valuable because it links algebraic procedures to geometric meaning. Gaussian elimination, for example, becomes more than a mechanical process for producing zeros: it reveals the structure of the system and helps identify its independent equations and possible solutions.
For students who have previously experienced linear algebra as a confusing collection of rules, this unified perspective can be transformative. It encourages them to recognise patterns, understand the relationships between concepts and develop a mental framework that remains useful when they encounter more advanced mathematics.
Geometric Intuition and Conceptual Understanding
Geometry is another central element of Strang’s teaching philosophy. Vectors, matrices and linear transformations are presented in ways that help readers visualise how mathematical operations affect space. Matrix multiplication can be understood as the composition of transformations, while eigenvectors reveal directions that remain unchanged under a transformation, apart from scaling.
This geometric interpretation gives meaning to calculations that might otherwise seem arbitrary. It also helps explain why certain mathematical tools are useful and how they relate to one another.
Strang’s approach is especially effective for students who learn by connecting symbolic manipulation with diagrams, examples and concrete problems. By developing intuition alongside computational skill, the book encourages readers to reason about mathematical structures rather than depend exclusively on memorised procedures.
Importantly, intuition does not mean avoiding difficult mathematics. Instead, it provides a foundation on which more formal reasoning can be built. Understanding the geometry of a problem often makes its algebraic properties easier to recognise and its solution easier to interpret.
Writing Style and Accessibility
One of the book’s most distinctive qualities is its conversational and personal style. Strang writes like an experienced teacher explaining difficult ideas to students, rather than an author attempting to establish authority through dense formalism. His explanations are direct, his examples are carefully chosen, and his enthusiasm for the subject is evident throughout.
This tone is particularly helpful for readers who find traditional mathematics textbooks intimidating. By introducing concepts through familiar computational problems and building towards greater abstraction, Strang lowers the initial barrier to entry without losing sight of the subject’s broader significance.
The approach also makes the book suitable for independent study. Readers can work through examples at their own pace, revisit challenging ideas and gradually develop confidence in their ability to solve problems. However, accessibility should not be confused with complete ease: mastering the material still requires sustained practice, careful reasoning and a willingness to work through unfamiliar concepts.
Applications and Practical Relevance
Strang’s treatment of applications is central to the book’s appeal. Linear algebra is connected to scientific computing, engineering models, graphs, networks, numerical algorithms and optimisation. These examples demonstrate how systems of equations and matrix operations can be used to solve problems that arise outside mathematics itself.
The book also explores numerical considerations, including computational methods and the limitations of calculations involving real data. Its discussions of linear programming, network problems and game theory illustrate how the subject extends into applied decision-making and operations research. The Mathematical Association of America’s review by Allen Stenger highlights the book’s coverage of numerical methods and its more detailed treatment of applications in networks and optimisation.
An important distinction, however, is that the book primarily teaches readers how linear algebra works and demonstrates where it can be applied; it does not turn them into specialists in every application area. Its examples provide a foundation for further study rather than a substitute for advanced training in numerical analysis, machine learning or engineering.
Nevertheless, this practical orientation gives the subject coherence and relevance. Students can see why mathematical techniques matter and how a solid understanding of linear algebra prepares them to tackle more advanced computational and scientific problems.
The Connection with MIT OpenCourseWare
The book’s educational value is enhanced by its association with Strang’s widely used MIT 18.06 linear algebra lectures, available through Open educational resources from the Massachusetts Institute of Technology. The lectures complement the textbook by presenting mathematical concepts through spoken explanations, worked examples and visual demonstrations.
This combination is particularly useful for independent learners. A student can read a section, watch the corresponding lecture, attempt the exercises and then revisit difficult concepts from a different perspective. The textbook provides a structured reference, while the lectures offer the experience of learning from an enthusiastic and experienced teacher.
Strang’s official linear algebra website also provides supplementary learning materials and links to the 18.06 lecture resources.
This synergy makes the book especially attractive to readers who do not have access to a university course or who want to reinforce material taught in lectures. It also illustrates how an effective textbook can become part of a broader learning environment rather than functioning as an isolated source of information.
Limitations and Critical Evaluation
Despite its many strengths, Linear Algebra and Its Applications has limitations, particularly for readers whose primary interest is theoretical mathematics.
The most significant criticism concerns the balance between intuition and formal proof. Strang frequently prioritises computational understanding, geometric explanations and applications over the systematic development of abstract theory. While this is highly effective for many applied students, it may leave readers less prepared for courses that demand precise theorem-proof arguments, detailed work with abstract vector spaces or a more axiomatic treatment of linear transformations.
This does not mean that the book lacks mathematical substance. Rather, its priorities differ from those of more theoretically oriented texts. Students intending to pursue pure mathematics may therefore wish to supplement it with a textbook that places greater emphasis on proof-writing and abstract structures.
A second limitation concerns the book’s age and its relationship to Strang’s later work. First published in 1976 and last updated in 2005, Linear Algebra and Its Applications remains valuable for its foundational material, but it has largely been superseded in Strang’s own teaching by Introduction to Linear Algebra. In its review of the earlier book, the Mathematical Association of America notes that the newer text follows a similar general approach while offering a more modern presentation and a more extensive treatment of singular value decomposition (SVD).
SVD is especially important in contemporary applications such as data analysis, dimensionality reduction and numerical computation. Readers whose main interests lie in these areas may therefore find Strang’s later text a more suitable starting point.
There is also a potential drawback to the conversational style itself. Some readers may prefer a more concise and formally structured exposition, with a clearer separation between definitions, theorems, proofs and examples. Strang’s approach rewards readers who enjoy exploring mathematical ideas through explanation and application, but it may be less suitable for those who want a highly systematic reference for formal results.
These limitations should be considered in relation to the book’s intended audience. Its purpose is to make linear algebra understandable and useful, not to provide the most abstract or comprehensive treatment possible.
Conclusion
Gilbert Strang’s Linear Algebra and Its Applications remains an influential and worthwhile textbook because it succeeds at something that is surprisingly difficult: making a fundamental mathematical subject understandable without reducing it to a collection of mechanical procedures.
Its greatest strengths are its geometric intuition, emphasis on the four fundamental subspaces, accessible writing style and sustained connection between mathematical theory and practical computation. By beginning with matrices and systems of equations, Strang gives students a concrete foundation from which they can develop a deeper understanding of vector spaces, orthogonality, eigenvalues and other essential concepts.
The book is particularly well suited to undergraduates in engineering, physics, computer science and applied mathematics, as well as independent learners who want to understand the subject rather than merely learn how to perform calculations. Its combination with MIT’s online lectures makes it even more useful as a self-study resource.
For students pursuing pure mathematics, supplementary reading will be necessary to develop stronger proof-writing skills and a more abstract understanding of linear algebra. Similarly, readers interested in the latest developments in data science and numerical methods may prefer Strang’s more recent textbooks.
Ultimately, the book’s enduring contribution is its insistence that computation, geometry and conceptual understanding belong together. It encourages students to see matrices not simply as arrays of numbers, but as representations of transformations, relationships and structures that appear throughout mathematics and the sciences.
Gilbert Strang’s Linear Algebra and Its Applications is a classic undergraduate textbook that presents linear algebra as a practical, intuitive and intellectually rewarding subject. Rather than beginning with abstract definitions and lengthy formal proofs, Strang develops the subject through concrete matrices, systems of equations and geometric interpretations. His approach reflects a central conviction: linear algebra becomes much easier to understand when students can see what mathematical operations mean, rather than simply memorising the rules for carrying them out.
The book is particularly valuable for students of mathematics, engineering, physics, computer science and other quantitative disciplines. It connects fundamental mathematical concepts with applications in scientific computing, network analysis and engineering, making it clear that linear algebra is not merely a collection of computational techniques but a framework for understanding relationships, transformations and complex systems.
Strang’s approachable writing style, emphasis on conceptual understanding and focus on practical applications have helped establish his textbooks as influential resources for generations of students. Nevertheless, the book’s accessible approach also raises questions about the balance between intuition, computation and formal mathematical rigour.
Content and Organisation
A defining feature of Strang’s approach is that he begins with concrete problems. Gaussian elimination, the solution of systems of linear equations and the structure of matrices provide an accessible entry point into the subject. These topics allow students to develop computational confidence before encountering more abstract ideas.
From this foundation, the book explores linear combinations, matrix multiplication, inverses, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and related concepts. The progression demonstrates how apparently separate techniques are connected through a common mathematical structure.
This organisation is particularly effective because it places computation and interpretation alongside one another. Students learn how to solve a system of equations while also considering when a solution exists, whether it is unique, and how the underlying equations determine the possible outcomes. Such questions encourage readers to look beyond numerical answers and understand the structure of the problems they are solving.
The book’s title is also significant. Its emphasis on applications distinguishes it from texts that concentrate primarily on abstract algebraic structures. Strang repeatedly demonstrates how linear algebra can be used to represent and analyse problems arising in science and engineering, giving the material a clear purpose beyond examination preparation.
The Four Fundamental Subspaces: The Book’s Central Strength
One of Strang’s most influential contributions to undergraduate linear algebra education is his explanation of the four fundamental subspaces of a matrix: the column space, row space, nullspace and left nullspace.
These spaces are often introduced as separate definitions, but Strang presents them as interconnected parts of a unified mathematical picture. The column space describes the possible outputs of a matrix transformation, while the nullspace identifies the inputs that the matrix maps to zero. The row space and left nullspace provide complementary information about the equations themselves and the relationships among their constraints.
This viewpoint helps students understand why systems of linear equations may have no solution, one solution or infinitely many solutions. It also clarifies the relationships between rank, linear independence and the dimensions of the relevant spaces.
The discussion is particularly valuable because it links algebraic procedures to geometric meaning. Gaussian elimination, for example, becomes more than a mechanical process for producing zeros: it reveals the structure of the system and helps identify its independent equations and possible solutions.
For students who have previously experienced linear algebra as a confusing collection of rules, this unified perspective can be transformative. It encourages them to recognise patterns, understand the relationships between concepts and develop a mental framework that remains useful when they encounter more advanced mathematics.
Geometric Intuition and Conceptual Understanding
Geometry is another central element of Strang’s teaching philosophy. Vectors, matrices and linear transformations are presented in ways that help readers visualise how mathematical operations affect space. Matrix multiplication can be understood as the composition of transformations, while eigenvectors reveal directions that remain unchanged under a transformation, apart from scaling.
This geometric interpretation gives meaning to calculations that might otherwise seem arbitrary. It also helps explain why certain mathematical tools are useful and how they relate to one another.
Strang’s approach is especially effective for students who learn by connecting symbolic manipulation with diagrams, examples and concrete problems. By developing intuition alongside computational skill, the book encourages readers to reason about mathematical structures rather than depend exclusively on memorised procedures.
Importantly, intuition does not mean avoiding difficult mathematics. Instead, it provides a foundation on which more formal reasoning can be built. Understanding the geometry of a problem often makes its algebraic properties easier to recognise and its solution easier to interpret.
Writing Style and Accessibility
One of the book’s most distinctive qualities is its conversational and personal style. Strang writes like an experienced teacher explaining difficult ideas to students, rather than an author attempting to establish authority through dense formalism. His explanations are direct, his examples are carefully chosen, and his enthusiasm for the subject is evident throughout.
This tone is particularly helpful for readers who find traditional mathematics textbooks intimidating. By introducing concepts through familiar computational problems and building towards greater abstraction, Strang lowers the initial barrier to entry without losing sight of the subject’s broader significance.
The approach also makes the book suitable for independent study. Readers can work through examples at their own pace, revisit challenging ideas and gradually develop confidence in their ability to solve problems. However, accessibility should not be confused with complete ease: mastering the material still requires sustained practice, careful reasoning and a willingness to work through unfamiliar concepts.
Applications and Practical Relevance
Strang’s treatment of applications is central to the book’s appeal. Linear algebra is connected to scientific computing, engineering models, graphs, networks, numerical algorithms and optimisation. These examples demonstrate how systems of equations and matrix operations can be used to solve problems that arise outside mathematics itself.
The book also explores numerical considerations, including computational methods and the limitations of calculations involving real data. Its discussions of linear programming, network problems and game theory illustrate how the subject extends into applied decision-making and operations research. The Mathematical Association of America’s review by Allen Stenger highlights the book’s coverage of numerical methods and its more detailed treatment of applications in networks and optimisation.
An important distinction, however, is that the book primarily teaches readers how linear algebra works and demonstrates where it can be applied; it does not turn them into specialists in every application area. Its examples provide a foundation for further study rather than a substitute for advanced training in numerical analysis, machine learning or engineering.
Nevertheless, this practical orientation gives the subject coherence and relevance. Students can see why mathematical techniques matter and how a solid understanding of linear algebra prepares them to tackle more advanced computational and scientific problems.
The Connection with MIT OpenCourseWare
The book’s educational value is enhanced by its association with Strang’s widely used MIT 18.06 linear algebra lectures, available through Open educational resources from the Massachusetts Institute of Technology. The lectures complement the textbook by presenting mathematical concepts through spoken explanations, worked examples and visual demonstrations.
This combination is particularly useful for independent learners. A student can read a section, watch the corresponding lecture, attempt the exercises and then revisit difficult concepts from a different perspective. The textbook provides a structured reference, while the lectures offer the experience of learning from an enthusiastic and experienced teacher.
Strang’s official linear algebra website also provides supplementary learning materials and links to the 18.06 lecture resources.
This synergy makes the book especially attractive to readers who do not have access to a university course or who want to reinforce material taught in lectures. It also illustrates how an effective textbook can become part of a broader learning environment rather than functioning as an isolated source of information.
Limitations and Critical Evaluation
Despite its many strengths, Linear Algebra and Its Applications has limitations, particularly for readers whose primary interest is theoretical mathematics.
The most significant criticism concerns the balance between intuition and formal proof. Strang frequently prioritises computational understanding, geometric explanations and applications over the systematic development of abstract theory. While this is highly effective for many applied students, it may leave readers less prepared for courses that demand precise theorem-proof arguments, detailed work with abstract vector spaces or a more axiomatic treatment of linear transformations.
This does not mean that the book lacks mathematical substance. Rather, its priorities differ from those of more theoretically oriented texts. Students intending to pursue pure mathematics may therefore wish to supplement it with a textbook that places greater emphasis on proof-writing and abstract structures.
A second limitation concerns the book’s age and its relationship to Strang’s later work. First published in 1976 and last updated in 2005, Linear Algebra and Its Applications remains valuable for its foundational material, but it has largely been superseded in Strang’s own teaching by Introduction to Linear Algebra. In its review of the earlier book, the Mathematical Association of America notes that the newer text follows a similar general approach while offering a more modern presentation and a more extensive treatment of singular value decomposition (SVD).
SVD is especially important in contemporary applications such as data analysis, dimensionality reduction and numerical computation. Readers whose main interests lie in these areas may therefore find Strang’s later text a more suitable starting point.
There is also a potential drawback to the conversational style itself. Some readers may prefer a more concise and formally structured exposition, with a clearer separation between definitions, theorems, proofs and examples. Strang’s approach rewards readers who enjoy exploring mathematical ideas through explanation and application, but it may be less suitable for those who want a highly systematic reference for formal results.
These limitations should be considered in relation to the book’s intended audience. Its purpose is to make linear algebra understandable and useful, not to provide the most abstract or comprehensive treatment possible.
Conclusion
Gilbert Strang’s Linear Algebra and Its Applications remains an influential and worthwhile textbook because it succeeds at something that is surprisingly difficult: making a fundamental mathematical subject understandable without reducing it to a collection of mechanical procedures.
Its greatest strengths are its geometric intuition, emphasis on the four fundamental subspaces, accessible writing style and sustained connection between mathematical theory and practical computation. By beginning with matrices and systems of equations, Strang gives students a concrete foundation from which they can develop a deeper understanding of vector spaces, orthogonality, eigenvalues and other essential concepts.
The book is particularly well suited to undergraduates in engineering, physics, computer science and applied mathematics, as well as independent learners who want to understand the subject rather than merely learn how to perform calculations. Its combination with MIT’s online lectures makes it even more useful as a self-study resource.
For students pursuing pure mathematics, supplementary reading will be necessary to develop stronger proof-writing skills and a more abstract understanding of linear algebra. Similarly, readers interested in the latest developments in data science and numerical methods may prefer Strang’s more recent textbooks.
Ultimately, the book’s enduring contribution is its insistence that computation, geometry and conceptual understanding belong together. It encourages students to see matrices not simply as arrays of numbers, but as representations of transformations, relationships and structures that appear throughout mathematics and the sciences.
February 19, 2016
Probably the best introduction to Linear Algebra out there.
I say this because
1. He focuses on why theorems are true and what they mean, rather than just grinding through the proofs.
At the end I felt I had gained insight as well as knowledge.
2. He gives enlightening examples of applications of Linear Algebra.
In many cases this gave me a new understanding of areas I had already covered.
Purists may not like it. His notation can be slightly sloppy. They can go and work through Linear Algebra Done Right if they wish.
Gilbert Strang's lectures on Linear Algebra on MIT Open Courseware are excellent and I recommend them.
Thank you Professor Strang for this wonderful book!
I say this because
1. He focuses on why theorems are true and what they mean, rather than just grinding through the proofs.
At the end I felt I had gained insight as well as knowledge.
2. He gives enlightening examples of applications of Linear Algebra.
In many cases this gave me a new understanding of areas I had already covered.
Purists may not like it. His notation can be slightly sloppy. They can go and work through Linear Algebra Done Right if they wish.
Gilbert Strang's lectures on Linear Algebra on MIT Open Courseware are excellent and I recommend them.
Thank you Professor Strang for this wonderful book!
January 16, 2022
Classic. More appropriate for applied scientists such as physicists or engineers.
The mathematicians will probably dislike it due to its emphasis in applications and the lack of proofs.
If you're a mathematician better look at Axler's Linear algebra done right.
The mathematicians will probably dislike it due to its emphasis in applications and the lack of proofs.
If you're a mathematician better look at Axler's Linear algebra done right.
February 7, 2015
A beginner book about linear algebra that tells you why linear algebra is like this.
January 18, 2016
Linear Algebra and its applications(Gilbert Strang)
2015. 10. 21
Symmetric Matrix = LDLT
differential Matrix는 Sysmmetric, Positive Definite
2015. 10. 22
Partial pivoting : 컴퓨터 연산시 큰 Pivot이 위로 올라가게 Permutation Matrix를 꾸며서 Gauss Elimination을 수행할때 |Lij|<1으로 설정하여 Roundoff Error를 줄인다.
2015. 10. 24
Vector Space의 정
1. x+y is in the subpsace
2. cx is inthe subspace
column space = C(A) = {b|Ax=b}
Null space = N(A) = {x|Ax=0}
2015. 10. 25
Retangular matrix (not square) can be converted to echelon form(staricase pattern, pivot이 Diagonal에 있지 않은 형태)
==> 더 정리하면 row reduce form으로 변경됨(쉽게 column, null Space를 찾아, complete solution을 찾을수 있다.)
2015. 10. 29
rank(AB)<=rank(A)
2015. 11. 4
Column Space : C(A) = r
null space : N(A) = n-r
Row Space : C(AT) = r , C(A)의 Dimension과 항상 동일하다
left null space : m-r
null space는 kernel of A로도 쓰인다
2015. 11. 10
Rank(A) = m ==> Right Inverse 존재(not unique) ==> 근이 많다 AT(AAT)-1
Rank(A) = n ==> Left Inverse 존재 ==> 근이 하나 (ATA)-1AT
if Rank(A) == 1 , A = uvT
2015. 11. 13
Vectors in the column space and left nullspace are perpendicular
2015. 11. 14
Graph 이론 Graph는 loop를 형성하며 모든 노드를 지나가는 선 Tree는 loop없이 모든 노드를 지나가는 선
Euler's formular : # nodes - #edges + # loop = 1
2015. 11. 19
Fundamental Equation : ATCAx=ATCb-f
2015. 12. 4
ATA = 대칭, A와 null space가 같다
2015. 12. 5
3D LeastSquare
function [m,p,s] = best_fit_line(x,y,z);
% x,y,z are n x 1 column vectors of the three coordinates
% of a set of n points in three dimensions. The best line,
% in the minimum mean square orthogonal distance sense,
% will pass through m and have direction cosines in p, so
% it can be expressed parametrically as x = m(1) + p(1)*t,
% y = m(2) + p(2)*t, and z = m(3)+p(3)*t, where t is the
% distance along the line from the mean point at m.
% s returns with the minimum mean square orthogonal
% distance to the line.
% RAS - March 14, 2005
[n,mx] = size(x); [ny,my] = size(y); [nz,mz] = size(z);
if (mx~=1)|(my~=1)|(mz~=1)|(ny~=n)|(nz~=n)
error('The arguments must be column vectors of the same length.')
end
m = [mean(x),mean(y),mean(z)];
w = [x-m(1),y-m(2),z-m(3)]; % Use "mean" point as base
a = (1/n)*w'*w; % 'a' is a positive definite matrix
[u,d,v] = svd(a); % 'eig' & 'svd' get same eigenvalues for this matrix
p = u(:,1)'; % Get eigenvector for largest eigenvalue
s = d(2,2)+d(3,3); % Sum the other two eigenvalues
2015. 12. 7
Weighted Least Square에서 error variance가 2일 때 weight는 1 이다
2015. 12. 8
Q(othogonal matrix)의 Transform에서는 길이가 변하지 않는다.
2015. 12. 15
Q Matrix Means Orthonormal columns
Qx=b, QTQx=QTb, x=QTb
A maxrix를 Gram schmit로 변환하고 나면 연산이 편하다.
QTQ = I 이므로 Least Square의 안정성이 증가한다
2015. 12. 24
Hilbert Space는 dimension이 무한대 이므로 함수도 포함된다.
Polynomial은 orthogonal 하지 않지만 -1
2015.12.28
determinant는 부피를 계산하는 것이 쓰일 수 있다.
2016.01.04
EigneValue를 구하는 공식은 없다.
det(A-I)가 5차 이상이면 즉 lambda^5의 근해 공식이 없다.
trace(A) = ()
det(A) = ()
eigenvalue로의 대각화 가능 여부(Eigenvector의 개수)와 역행렬 가능 여부(eigenvalue가 0이 없음)는 다르다.
AB=BA인 경우 A와 B의 eigenvector는 같다.
markove matrix는 각 column의 합이 1, 음수 항은 없다.
⇒ A markov matrix has all aij>0, with each column adding to 1.
positive matrix (not positive definite matrix) has all aij>0
⇒ eigenvector of largest eigenvalue is positive.
2016.01.05
미방을 푸는데 Eigenvalue가 중근이면 generalized eigenvector가 도입되어야 하고 이런 경우가 Jordan Form으로 나타난다
If A is skew-sysmmetric(AT = -A) then eAt is an orthogonal matrix
2016.01.06
Every sysmmetric matrix(Hermitian matrix) has real eigenvalue.
Its eigenvectors can be chosen to be orthonormal
A가 sysmmetric하여 A=QQT로 변환되면 spectral theorem이라 한다.
여기서 Q는 orthonomal한 Eigenvector로 특정축으로 투영된 값이 에 반영되므로 저렇게 명명된다.
2016.01.12
Symmetric, skew-sysmmetric, orthogonal, Hermition, skew Hermition, and unitary matrices are called “normal matrix”.
Normal matrices are exactly those that have a complete set of orthonomal eigenvectos. ⇒ Diagonalizable
2016.01.15
함수 F의 Taylor series는 아래와 같다.
F(x) = F(0) +xT(grad F)+12xTAx+higer order terms
F의 최대 최소를 찾기 위해서는 Taylor series의 2차항만을 확인하여도 된다.
즉 A의 positive(negative) definite를 확인하면 된다.
2016.01.18
Positive Definite Test 방안
(1)xTAx>0for all nonzero real vectors x.
(2) All the eigenvalues of A satisfy i>0
(3) All the upper left submatrices Ak have positive determinants.
(4) All the pivots (without row exchanges) satisfy dk>0
2016.1.19
Congruence transformation A→ CTAC for some nonsingular C
CTAC has the same number of positive eigenvalues, negtive eigenvalues, and zero eigenvalues as A.
For any sysmmetric matrix A=LDLT은 Pivots의 부호와 eigenvalue의 부호가 같다.
⇒ eigenvalue 찾는 방법으로 쓸 수 있다.
Mu''+Au=0⇒ Ax=Mx⇒ RTRx⇒ AR-1y=Ry
2016.1.21
Lagrange multipliers.
P(x) = 12xTAx-xTb, Cx=d ( Constraint ) ⇒ 문제를 풀기 위해 y를 도입한다.
L(x,y) = 12xTAx-xTb+yT(Cx-d)
L(x,y)를 x와 y에 대해서 미분하면 x와 y가 구해진다. ( P(x)와 Cx최소가 되는 것은 둘의 방정식이 하나의 접점을 갖는 경우 즉 L(x,y)=0,둘의 접선의 기울기가 같은 경우 )
minimum = 112bTAb+yT(Cx-d)
Rayleigh Quotient
R(x1)=x1TAx1x1Tx1=x1Tx1x1Tx1
minR(x)max
2016.01.25
컴퓨터로 Matrix 연산 시 자릿수에 의한 오차 분석
최대 오차를 Condition Number(c)라고 부른다.
Matrix가 symmetric하면 c = max/min
Matrix가 unsymmetric하면 c = ||A||||A-1|| ||A|| = max||Ax||||x||
Matrix Norm
The norm of A measures the largest amount by which any vector is amplified by matrix multiplication : ||A|| = max||Ax||||x||
2015. 10. 21
Symmetric Matrix = LDLT
differential Matrix는 Sysmmetric, Positive Definite
2015. 10. 22
Partial pivoting : 컴퓨터 연산시 큰 Pivot이 위로 올라가게 Permutation Matrix를 꾸며서 Gauss Elimination을 수행할때 |Lij|<1으로 설정하여 Roundoff Error를 줄인다.
2015. 10. 24
Vector Space의 정
1. x+y is in the subpsace
2. cx is inthe subspace
column space = C(A) = {b|Ax=b}
Null space = N(A) = {x|Ax=0}
2015. 10. 25
Retangular matrix (not square) can be converted to echelon form(staricase pattern, pivot이 Diagonal에 있지 않은 형태)
==> 더 정리하면 row reduce form으로 변경됨(쉽게 column, null Space를 찾아, complete solution을 찾을수 있다.)
2015. 10. 29
rank(AB)<=rank(A)
2015. 11. 4
Column Space : C(A) = r
null space : N(A) = n-r
Row Space : C(AT) = r , C(A)의 Dimension과 항상 동일하다
left null space : m-r
null space는 kernel of A로도 쓰인다
2015. 11. 10
Rank(A) = m ==> Right Inverse 존재(not unique) ==> 근이 많다 AT(AAT)-1
Rank(A) = n ==> Left Inverse 존재 ==> 근이 하나 (ATA)-1AT
if Rank(A) == 1 , A = uvT
2015. 11. 13
Vectors in the column space and left nullspace are perpendicular
2015. 11. 14
Graph 이론 Graph는 loop를 형성하며 모든 노드를 지나가는 선 Tree는 loop없이 모든 노드를 지나가는 선
Euler's formular : # nodes - #edges + # loop = 1
2015. 11. 19
Fundamental Equation : ATCAx=ATCb-f
2015. 12. 4
ATA = 대칭, A와 null space가 같다
2015. 12. 5
3D LeastSquare
function [m,p,s] = best_fit_line(x,y,z);
% x,y,z are n x 1 column vectors of the three coordinates
% of a set of n points in three dimensions. The best line,
% in the minimum mean square orthogonal distance sense,
% will pass through m and have direction cosines in p, so
% it can be expressed parametrically as x = m(1) + p(1)*t,
% y = m(2) + p(2)*t, and z = m(3)+p(3)*t, where t is the
% distance along the line from the mean point at m.
% s returns with the minimum mean square orthogonal
% distance to the line.
% RAS - March 14, 2005
[n,mx] = size(x); [ny,my] = size(y); [nz,mz] = size(z);
if (mx~=1)|(my~=1)|(mz~=1)|(ny~=n)|(nz~=n)
error('The arguments must be column vectors of the same length.')
end
m = [mean(x),mean(y),mean(z)];
w = [x-m(1),y-m(2),z-m(3)]; % Use "mean" point as base
a = (1/n)*w'*w; % 'a' is a positive definite matrix
[u,d,v] = svd(a); % 'eig' & 'svd' get same eigenvalues for this matrix
p = u(:,1)'; % Get eigenvector for largest eigenvalue
s = d(2,2)+d(3,3); % Sum the other two eigenvalues
2015. 12. 7
Weighted Least Square에서 error variance가 2일 때 weight는 1 이다
2015. 12. 8
Q(othogonal matrix)의 Transform에서는 길이가 변하지 않는다.
2015. 12. 15
Q Matrix Means Orthonormal columns
Qx=b, QTQx=QTb, x=QTb
A maxrix를 Gram schmit로 변환하고 나면 연산이 편하다.
QTQ = I 이므로 Least Square의 안정성이 증가한다
2015. 12. 24
Hilbert Space는 dimension이 무한대 이므로 함수도 포함된다.
Polynomial은 orthogonal 하지 않지만 -1
2015.12.28
determinant는 부피를 계산하는 것이 쓰일 수 있다.
2016.01.04
EigneValue를 구하는 공식은 없다.
det(A-I)가 5차 이상이면 즉 lambda^5의 근해 공식이 없다.
trace(A) = ()
det(A) = ()
eigenvalue로의 대각화 가능 여부(Eigenvector의 개수)와 역행렬 가능 여부(eigenvalue가 0이 없음)는 다르다.
AB=BA인 경우 A와 B의 eigenvector는 같다.
markove matrix는 각 column의 합이 1, 음수 항은 없다.
⇒ A markov matrix has all aij>0, with each column adding to 1.
positive matrix (not positive definite matrix) has all aij>0
⇒ eigenvector of largest eigenvalue is positive.
2016.01.05
미방을 푸는데 Eigenvalue가 중근이면 generalized eigenvector가 도입되어야 하고 이런 경우가 Jordan Form으로 나타난다
If A is skew-sysmmetric(AT = -A) then eAt is an orthogonal matrix
2016.01.06
Every sysmmetric matrix(Hermitian matrix) has real eigenvalue.
Its eigenvectors can be chosen to be orthonormal
A가 sysmmetric하여 A=QQT로 변환되면 spectral theorem이라 한다.
여기서 Q는 orthonomal한 Eigenvector로 특정축으로 투영된 값이 에 반영되므로 저렇게 명명된다.
2016.01.12
Symmetric, skew-sysmmetric, orthogonal, Hermition, skew Hermition, and unitary matrices are called “normal matrix”.
Normal matrices are exactly those that have a complete set of orthonomal eigenvectos. ⇒ Diagonalizable
2016.01.15
함수 F의 Taylor series는 아래와 같다.
F(x) = F(0) +xT(grad F)+12xTAx+higer order terms
F의 최대 최소를 찾기 위해서는 Taylor series의 2차항만을 확인하여도 된다.
즉 A의 positive(negative) definite를 확인하면 된다.
2016.01.18
Positive Definite Test 방안
(1)xTAx>0for all nonzero real vectors x.
(2) All the eigenvalues of A satisfy i>0
(3) All the upper left submatrices Ak have positive determinants.
(4) All the pivots (without row exchanges) satisfy dk>0
2016.1.19
Congruence transformation A→ CTAC for some nonsingular C
CTAC has the same number of positive eigenvalues, negtive eigenvalues, and zero eigenvalues as A.
For any sysmmetric matrix A=LDLT은 Pivots의 부호와 eigenvalue의 부호가 같다.
⇒ eigenvalue 찾는 방법으로 쓸 수 있다.
Mu''+Au=0⇒ Ax=Mx⇒ RTRx⇒ AR-1y=Ry
2016.1.21
Lagrange multipliers.
P(x) = 12xTAx-xTb, Cx=d ( Constraint ) ⇒ 문제를 풀기 위해 y를 도입한다.
L(x,y) = 12xTAx-xTb+yT(Cx-d)
L(x,y)를 x와 y에 대해서 미분하면 x와 y가 구해진다. ( P(x)와 Cx최소가 되는 것은 둘의 방정식이 하나의 접점을 갖는 경우 즉 L(x,y)=0,둘의 접선의 기울기가 같은 경우 )
minimum = 112bTAb+yT(Cx-d)
Rayleigh Quotient
R(x1)=x1TAx1x1Tx1=x1Tx1x1Tx1
minR(x)max
2016.01.25
컴퓨터로 Matrix 연산 시 자릿수에 의한 오차 분석
최대 오차를 Condition Number(c)라고 부른다.
Matrix가 symmetric하면 c = max/min
Matrix가 unsymmetric하면 c = ||A||||A-1|| ||A|| = max||Ax||||x||
Matrix Norm
The norm of A measures the largest amount by which any vector is amplified by matrix multiplication : ||A|| = max||Ax||||x||
This entire review has been hidden because of spoilers.
January 4, 2020
It's hard to mark this book as ever "done". We covered chapters of this as part of a reading club and I highly recommend doing the same. If you're interested in starting a career in machine learning or scientific computing, this is a must have on your shelf. Even if you're familiar with his video lectures, the book is worth reading as it tends to solidify the concepts. One can never say they are experts in the topics in the book. Every time I read (or re-read) some page in the book, I learn something new. It's one for the classics!
April 19, 2020
I took MIT course on Linear Algebra by Gilbert Strang and complimented my learning with his textbook, I must say I wish I have had this on my lectures at university, you understand so much , from the very base and basic til the complex topics on Linear Algebra , it covers a wide range of topics needed for Machine Learning and Computing in General. Strongly recommend this text if you are interested to get a grasp on the basics.
January 2, 2021
The author Gilbert Strang is an authoritative figure on the subject and I am fortunate to have studied his MIT ocw course...his textbook is an excellent introduction to the subject of linear algebra with an attempt to understand the subject intuitively and visually. The different perspectives that he introduces are essential to get the essence of the subject. It's an excellent read alongside his excellent course.
December 17, 2023
While I was studying math, linear algebra was my one of favorite subjects and after I've switched to research physics it became even more so. This book has its permanent place on my work desk alongside Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz and good 'ol Matematički priručnik. For a good reason. This is one of the best and most comprehensive books on the subject. And linear algebra is one of the greatest and most powerful tools we have for solving problems.
"Mathematics is the art of reducing any problem to linear algebra." - William Stein
"If you can reduce a mathematical problem to a problem in linear algebra, you can most likely solve it, provided you know enough linear algebra." - Peter D. Lax
I'll also leave this amazing video by Michael Penn
The unreasonable effectiveness of Linear Algebra
"Mathematics is the art of reducing any problem to linear algebra." - William Stein
"If you can reduce a mathematical problem to a problem in linear algebra, you can most likely solve it, provided you know enough linear algebra." - Peter D. Lax
I'll also leave this amazing video by Michael Penn
The unreasonable effectiveness of Linear Algebra
Read
April 27, 2024When I was looking for a linear algebra book, something I could teach myself from on my own, I was a bit hesitant what to pick. After chapter 1 and 2, I was still not comfortable with the style and there were several things I couldn't understand even after reading everything multiple and multiple times. Fortunately, I decided to go on with this book. And it was after chapter 4 that I started getting comfotable with the teaching style of the book. I'd say the part on linear transformation is still a bit messy. I had lots of problem following things there. Overall, however, I am really glad I kept reading this book, because even though it might not seem so at first, I think the topcis and the way they've been organized is just better than so many other textbooks I'd seen, neither too simple nor too complicated and detailed. And it covers basic linear algebra one would need in a beautiful way. Great read.
April 23, 2022
I don’t get why this book gets recommended so often. I didn’t find the explanations particularly easy to follow (compared to a book that doesn’t necessarily want to be completely rigorous everywhere), so the only appeal that might have for some is that it lists a fair amount of applications.
On the other hand one thing I really liked is that it introduced (left and right) column and null spaces early, and the relation between their dimensions and the size of the matrix.
On the other hand one thing I really liked is that it introduced (left and right) column and null spaces early, and the relation between their dimensions and the size of the matrix.
October 16, 2024
Κορυφαίο. Γραμμένο με απλό τρόπο για να διευκολύνει την κατανόηση της θεωρίας, δεν παύει όμως να πραγματεύεται μια δύσκολη θεματολογία, επομένως χρειάζεται προσεκτική μελέτη. Επεξηγηματικό και εμβριθές· δίδει περισσότερο βάρος στην σκέψη πίσω από την απόδειξη παρά σε αυτήν καθαυτή. Έτσι, το παρόν βιβλίο σφυρηλατεί τη διαίσθηση του αναγνώστη περί της θεωρίας. Πραγματικά πολύ καλή προσέγγιση και βιβλίο.
January 3, 2018
The best Linear Algebra text book of this century. Written by a true master of the subject and explained in a way that even non-mathematicans can understand. I would recommend this book to anyone start out studying the subject of linear algebra. This book combined with his MIT open course ware lectures proves a fantastic approach to the subject.
February 14, 2019
This was my first major text in linear algebra, although I already knew some basic properties of matrices. I have no complaints about this book, it seems to do a good job explaining all the concepts and it's easy to follow assuming you don't skip around and make a conscious effort to understand beyond reading.
May 25, 2020
I have finished my Linear algebra course and got an A. I do admitted that I don't finished the entire book. If you have some basic knowledge then this book is good for you to contribute your Matrix knowledge. But it not suit for those who want to learn the knowledge of Linear algebra for the first time.
March 7, 2026
Strang is the GOAT of Linear Algebra for a reason. Bro is funny, explains things in a clear and concise manner, and his exercises are pretty helpful. I do wish he talked a little bit more about change-of-basis and change-of-coordinate matrices, but other than that I really enjoyed this textbook as an introduction to Linear Algebra.
November 3, 2026
Quick take: Linear Algebra and Its Applications is a clever, crowd-pleasing page-turner that lives and dies by its hook. What worked: a killer premise and chapters short enough to read on a coffee break. What didn't: the characters are thin, and the dialogue sometimes sounds like a script. Recommend for readers who just want a fun ride, skip for anyone who wants to feel anything deep.
May 11, 2020
Excellent book and Excellent lectures by Prof. Strang. Explains Linear Algebra in a very lucid manner with lot of examples. Also, this books give so many applications of Linear Algebra in different fields which works as motivation to explore further.
June 12, 2021
Probably the best reading experience of any Gilbert Strang book, and I say that having tried a few; not like his new books where the author can't seem to write in full sentences, but not so old that the applications no longer make sense; overall, a good first linear algebra book.
March 12, 2024
THE best book on linear algebra! The book explains concepts in a very comprehensive way, is very understable and informative. This is probably your first choice if you want to self study linear algebra, although it may lack drills, and you will have to refer to other materials for them
December 4, 2017
Thoroughly entertaining.
I was thrust into using linear algebra without any formal education in it. Strang explains very well why it works the way it does.
Goes well with his OCW scholar course.
I was thrust into using linear algebra without any formal education in it. Strang explains very well why it works the way it does.
Goes well with his OCW scholar course.
September 9, 2018
Absolutely great textbook from a great professor. The greek translation of the book is spot-on as well.
May 26, 2019
A bit hard to understand at first so I moved to another books and courses before coming back. The content itself is amazing !
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