https://mklab.gr/ Sat, 12 Sep 2026 21:55:25 +0000 MyBB https://mklab.gr/showthread.php?tid=1823 Fri, 04 Sep 2026 01:37:27 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1823
Applied Linear Algebra
Authors: Peter J. Olver, Chehrzad Shakiban
Publication date: 30 May 2018 (eBook)
Publisher: Springer Cham
Edition: 2nd edition
Series:Undergraduate Texts in Mathematics

Applied Linear Algebra is an undergraduate textbook that develops the fundamental ideas and techniques of linear algebra while consistently connecting them to applications in science, engineering, computing, numerical analysis, data analysis, and dynamical systems. Rather than treating linear algebra purely abstractly, Olver and Shakiban emphasize the interaction between theory and practical problems: Gaussian elimination leads naturally to solving linear systems, inner products and orthogonality lead to approximation and least-squares methods, while eigenvalues and singular values provide tools for understanding transformations, stability, data reduction, and dynamical behaviour.

The book progresses from linear systems, vector spaces and bases through inner products, norms, orthogonality, minimization and least squares, before developing more advanced topics such as linear transformations, eigenvalues, singular values, iterative methods and dynamical systems. The second edition expands applications involving numerical methods, signal processing and data analysis while improving the presentation of the theoretical material. Particularly important is the book's effort to explain why linear-algebraic techniques work and when they should be used, rather than presenting them merely as computational algorithms. 

Only single-variable calculus is formally required, and no previous course in linear algebra is assumed, although increasing mathematical maturity is helpful as the material becomes more abstract. It can therefore serve either as a substantial first course in linear algebra or as a more application-oriented second course. Its treatment also provides a strong foundation for later work in differential equations, numerical analysis, statistics and data science. 

Key takeaways

• Strong balance between theoretical linear algebra and real applications.
  
• Covers both classical material—matrices, vector spaces, orthogonality and eigenvalues—and modern topics such as SVD, least squares, iteration, data analysis and dynamical systems. 
  
• Particularly suitable for students interested in applied mathematics, engineering, scientific computing or data science.
  
• More comprehensive than a typical introductory linear-algebra text, reaching naturally toward numerical and applied mathematics.
  

BOOK]]>
Applied Linear Algebra
Authors: Peter J. Olver, Chehrzad Shakiban
Publication date: 30 May 2018 (eBook)
Publisher: Springer Cham
Edition: 2nd edition
Series:Undergraduate Texts in Mathematics

Applied Linear Algebra is an undergraduate textbook that develops the fundamental ideas and techniques of linear algebra while consistently connecting them to applications in science, engineering, computing, numerical analysis, data analysis, and dynamical systems. Rather than treating linear algebra purely abstractly, Olver and Shakiban emphasize the interaction between theory and practical problems: Gaussian elimination leads naturally to solving linear systems, inner products and orthogonality lead to approximation and least-squares methods, while eigenvalues and singular values provide tools for understanding transformations, stability, data reduction, and dynamical behaviour.

The book progresses from linear systems, vector spaces and bases through inner products, norms, orthogonality, minimization and least squares, before developing more advanced topics such as linear transformations, eigenvalues, singular values, iterative methods and dynamical systems. The second edition expands applications involving numerical methods, signal processing and data analysis while improving the presentation of the theoretical material. Particularly important is the book's effort to explain why linear-algebraic techniques work and when they should be used, rather than presenting them merely as computational algorithms. 

Only single-variable calculus is formally required, and no previous course in linear algebra is assumed, although increasing mathematical maturity is helpful as the material becomes more abstract. It can therefore serve either as a substantial first course in linear algebra or as a more application-oriented second course. Its treatment also provides a strong foundation for later work in differential equations, numerical analysis, statistics and data science. 

Key takeaways

• Strong balance between theoretical linear algebra and real applications.
  
• Covers both classical material—matrices, vector spaces, orthogonality and eigenvalues—and modern topics such as SVD, least squares, iteration, data analysis and dynamical systems. 
  
• Particularly suitable for students interested in applied mathematics, engineering, scientific computing or data science.
  
• More comprehensive than a typical introductory linear-algebra text, reaching naturally toward numerical and applied mathematics.
  

BOOK]]> https://mklab.gr/showthread.php?tid=1671 Mon, 17 Aug 2026 19:48:15 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1671 Matrix Algebra: Theory, Computations, and Applications in Statistics
Author: James E. Gentle
First published: 2007
Publisher: Springer
Series: Springer Texts in Statistics

Matrix Algebra: Theory, Computations, and Applications in Statistics is a comprehensive treatment of matrix and linear algebra written specifically with statistics and data analysis in mind. Gentle develops the subject from vectors, vector spaces, and fundamental matrix properties through linear systems, eigenvalues and eigenvectors, matrix transformations, and factorizations. Rather than treating these topics purely abstractly, the book continually emphasizes why particular matrix concepts matter in statistical work. Important classes of matrices—including projection, positive-definite, and other structured matrices—receive special attention because of their central role in statistical models. 

A particularly valuable feature is the connection between mathematical theory and computation. Gentle does not stop after establishing algebraic results; he examines how matrix calculations are actually performed numerically. This includes numerical computation, matrix factorizations, solving systems of linear equations, and algorithms for calculating eigenvalues and eigenvectors. Consequently, the book occupies an interesting position between a theoretical linear-algebra textbook and a numerical/computational statistics text. This makes it especially useful for readers interested in statistics, machine learning, data science, econometrics, or scientific computing rather than linear algebra solely as a pure mathematical subject. 

The book is best suited to advanced undergraduate or graduate students and researchers who need matrix algebra as a working mathematical tool. It can be demanding as a first encounter with linear algebra, but for someone with basic mathematical maturity it offers considerable depth and can serve as a long-term reference. Its continuing relevance is reflected in subsequent editions: Springer published a substantially expanded second edition in 2017, while the third edition appeared in 2024, adding greater emphasis on R, statistical linear models, statistics, and data science. 

Key takeaways

• Statistics-oriented: matrix algebra is developed around applications in statistics and data analysis.
  
• Theory + computation: combines rigorous mathematical foundations with numerical methods and algorithms.
  
• Broad coverage: includes vector spaces, matrix properties, factorizations, linear systems, eigenanalysis, special matrices, and statistical applications.
  
• Excellent reference: particularly valuable for graduate students and researchers in statistics, data science and computational mathematics.
  

Springer — Matrix Algebra by James E. Gentle]]> Matrix Algebra: Theory, Computations, and Applications in Statistics
Author: James E. Gentle
First published: 2007
Publisher: Springer
Series: Springer Texts in Statistics

Matrix Algebra: Theory, Computations, and Applications in Statistics is a comprehensive treatment of matrix and linear algebra written specifically with statistics and data analysis in mind. Gentle develops the subject from vectors, vector spaces, and fundamental matrix properties through linear systems, eigenvalues and eigenvectors, matrix transformations, and factorizations. Rather than treating these topics purely abstractly, the book continually emphasizes why particular matrix concepts matter in statistical work. Important classes of matrices—including projection, positive-definite, and other structured matrices—receive special attention because of their central role in statistical models. 

A particularly valuable feature is the connection between mathematical theory and computation. Gentle does not stop after establishing algebraic results; he examines how matrix calculations are actually performed numerically. This includes numerical computation, matrix factorizations, solving systems of linear equations, and algorithms for calculating eigenvalues and eigenvectors. Consequently, the book occupies an interesting position between a theoretical linear-algebra textbook and a numerical/computational statistics text. This makes it especially useful for readers interested in statistics, machine learning, data science, econometrics, or scientific computing rather than linear algebra solely as a pure mathematical subject. 

The book is best suited to advanced undergraduate or graduate students and researchers who need matrix algebra as a working mathematical tool. It can be demanding as a first encounter with linear algebra, but for someone with basic mathematical maturity it offers considerable depth and can serve as a long-term reference. Its continuing relevance is reflected in subsequent editions: Springer published a substantially expanded second edition in 2017, while the third edition appeared in 2024, adding greater emphasis on R, statistical linear models, statistics, and data science. 

Key takeaways

• Statistics-oriented: matrix algebra is developed around applications in statistics and data analysis.
  
• Theory + computation: combines rigorous mathematical foundations with numerical methods and algorithms.
  
• Broad coverage: includes vector spaces, matrix properties, factorizations, linear systems, eigenanalysis, special matrices, and statistical applications.
  
• Excellent reference: particularly valuable for graduate students and researchers in statistics, data science and computational mathematics.
  

Springer — Matrix Algebra by James E. Gentle]]> https://mklab.gr/showthread.php?tid=1669 Mon, 17 Aug 2026 19:40:59 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1669 Book:Linear Algebra
Author: Serge Lang
First published: 1970
Series:Undergraduate Texts in Mathematics

Serge Lang’s Linear Algebra is a compact, rigorous treatment intended primarily for students who already have some familiarity with matrices, determinants, and elementary linear transformations. Rather than treating linear algebra mainly as a collection of computational techniques, Lang develops it as an abstract mathematical theory centered on vector spaces and linear maps. The book moves from the fundamental structure of vector spaces to eigenvectors and eigenvalues, quadratic and Hermitian forms, and the diagonalization of symmetric, Hermitian, and unitary operators. It also develops more advanced structural results, including triangularization and the Jordan canonical form. Although Lang assumes mathematical maturity, the exposition is logically self-contained. 

A distinctive feature is the book's emphasis on proofs and mathematical structure. Lang is interested not merely in how to manipulate matrices but in explaining why the main results of linear algebra work and how apparently different ideas fit together. Some proofs are particularly elegant—for example, Goodreads reviewers highlight alternative arguments for results such as the Cayley–Hamilton theorem and the existence of eigenvectors for symmetric matrices. The book also goes somewhat beyond a conventional undergraduate syllabus with material on convex sets and the finite-dimensional Krein–Milman theorem. Exercises form an important part of the presentation, reinforcing both technique and theoretical understanding. 

This is therefore better viewed as a second or more mathematically sophisticated encounter with linear algebra than as a gentle first introduction. Students wanting extensive numerical examples and step-by-step matrix calculations may find Lang terse, whereas readers interested in abstract algebra, analysis, geometry, or higher mathematics are likely to appreciate his concise and structural approach. It is especially useful preparation for subjects where vector spaces and linear operators become part of the underlying language—functional analysis, differential equations, representation theory, numerical mathematics, and advanced geometry.

Key Takeaways

• Structure over computation: the emphasis is on vector spaces, linear maps, invariants, and proofs rather than simply matrix calculation.
  
• Substantial theoretical coverage: eigenvalues, diagonalization, quadratic and Hermitian forms, triangularization, Jordan form, and convexity all receive attention. 
  
• Best for mathematically mature readers: some previous exposure to elementary linear algebra is helpful even though the text is formally self-contained. 
  
• Excellent bridge to higher mathematics: Lang presents linear algebra as a fundamental mathematical language rather than an isolated computational subject.
  

BOOK]]> Book:Linear Algebra
Author: Serge Lang
First published: 1970
Series:Undergraduate Texts in Mathematics

Serge Lang’s Linear Algebra is a compact, rigorous treatment intended primarily for students who already have some familiarity with matrices, determinants, and elementary linear transformations. Rather than treating linear algebra mainly as a collection of computational techniques, Lang develops it as an abstract mathematical theory centered on vector spaces and linear maps. The book moves from the fundamental structure of vector spaces to eigenvectors and eigenvalues, quadratic and Hermitian forms, and the diagonalization of symmetric, Hermitian, and unitary operators. It also develops more advanced structural results, including triangularization and the Jordan canonical form. Although Lang assumes mathematical maturity, the exposition is logically self-contained. 

A distinctive feature is the book's emphasis on proofs and mathematical structure. Lang is interested not merely in how to manipulate matrices but in explaining why the main results of linear algebra work and how apparently different ideas fit together. Some proofs are particularly elegant—for example, Goodreads reviewers highlight alternative arguments for results such as the Cayley–Hamilton theorem and the existence of eigenvectors for symmetric matrices. The book also goes somewhat beyond a conventional undergraduate syllabus with material on convex sets and the finite-dimensional Krein–Milman theorem. Exercises form an important part of the presentation, reinforcing both technique and theoretical understanding. 

This is therefore better viewed as a second or more mathematically sophisticated encounter with linear algebra than as a gentle first introduction. Students wanting extensive numerical examples and step-by-step matrix calculations may find Lang terse, whereas readers interested in abstract algebra, analysis, geometry, or higher mathematics are likely to appreciate his concise and structural approach. It is especially useful preparation for subjects where vector spaces and linear operators become part of the underlying language—functional analysis, differential equations, representation theory, numerical mathematics, and advanced geometry.

Key Takeaways

• Structure over computation: the emphasis is on vector spaces, linear maps, invariants, and proofs rather than simply matrix calculation.
  
• Substantial theoretical coverage: eigenvalues, diagonalization, quadratic and Hermitian forms, triangularization, Jordan form, and convexity all receive attention. 
  
• Best for mathematically mature readers: some previous exposure to elementary linear algebra is helpful even though the text is formally self-contained. 
  
• Excellent bridge to higher mathematics: Lang presents linear algebra as a fundamental mathematical language rather than an isolated computational subject.
  

BOOK]]> https://mklab.gr/showthread.php?tid=1654 Mon, 17 Aug 2026 18:58:40 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1654 Author: Steven Roman
Publication: 3rd edition, 2008
Publisher: Springer New York
Series:Graduate Texts in Mathematics, Vol. 135

Steven Roman’s Advanced Linear Algebra is a broad and rigorous treatment of linear algebra that goes substantially beyond the standard undergraduate course. Rather than concentrating mainly on matrix calculations, Roman develops linear algebra from a structural and algebraic perspective. The book begins with vector spaces, linear transformations, quotient spaces and the isomorphism theorems, but moves relatively quickly through these foundations. It then introduces modules, deliberately emphasizing their relationship with vector spaces. Modules over principal ideal domains provide the machinery for understanding the structure of linear operators and lead naturally to canonical forms, eigenvalues and eigenvectors. 

A major strength of the book is the way it connects the algebraic and geometric sides of the subject. Roman develops real and complex inner-product spaces and the structure theory of normal operators, culminating in finite-dimensional spectral theory. From there, the scope becomes unusually wide: bilinear forms, metric spaces, Hilbert spaces, tensor products, convexity and separation, affine geometry, singular values, and the Moore–Penrose inverse all receive attention. The third edition also contains a chapter on associative algebras, including Frobenius's characterization of finite-dimensional real division algebras and Wedderburn's theorem for finite division algebras, as well as material on the spectral mapping theorem.

This breadth makes Advanced Linear Algebra closer to a comprehensive reference work than a conventional linear-algebra textbook. Particularly valuable is Roman's treatment of topics that students often encounter separately in abstract algebra, functional analysis, geometry, or numerical linear algebra. Tensor products, Hilbert spaces, canonical forms, normal operators and associative algebras are presented as parts of a larger mathematical structure rather than as unrelated techniques. The final treatment of the umbral calculus is especially unusual for a linear-algebra textbook and illustrates just how far Roman is willing to extend the subject. 

The price of this breadth is difficulty. Although Roman develops the necessary basic linear algebra, the presentation is rapid, and Springer describes prior linear algebra plus a degree of mathematical maturity as highly desirable.  This is therefore not the ideal first encounter with linear algebra. It is much better suited to someone who already understands matrices, vector spaces, bases, linear transformations, eigenvalues and elementary proofs and now wants to understand why the subject has the structure it does. The exercises and proof-oriented presentation also make it particularly appropriate for graduate study and preparation for more advanced work in algebra, geometry and analysis.

Key Takeaways

• Structural rather than computational: the emphasis is on vector spaces, transformations, modules, operators and proofs rather than routine matrix calculations.
  
• Exceptionally broad: it progresses from classical linear algebra to modules, spectral theory, Hilbert spaces, tensor products, convexity, affine geometry, generalized inverses and associative algebras. 
  
• Best for experienced readers: some previous linear algebra and mathematical maturity are strongly recommended.
  
• Excellent reference: its breadth makes it useful well beyond a single graduate course; reviewers cited by Springer specifically praise its comprehensive coverage, proofs and exercises. 
  
  

Overall:Advanced Linear Algebra is an excellent choice for a reader who wants to make the transition from elementary linear algebra to the more abstract viewpoint used in modern mathematics. It is demanding, but its combination of depth, breadth and structural insight makes it particularly valuable as both a graduate textbook and a long-term mathematical reference.


BOOK]]> Author: Steven Roman
Publication: 3rd edition, 2008
Publisher: Springer New York
Series:Graduate Texts in Mathematics, Vol. 135

Steven Roman’s Advanced Linear Algebra is a broad and rigorous treatment of linear algebra that goes substantially beyond the standard undergraduate course. Rather than concentrating mainly on matrix calculations, Roman develops linear algebra from a structural and algebraic perspective. The book begins with vector spaces, linear transformations, quotient spaces and the isomorphism theorems, but moves relatively quickly through these foundations. It then introduces modules, deliberately emphasizing their relationship with vector spaces. Modules over principal ideal domains provide the machinery for understanding the structure of linear operators and lead naturally to canonical forms, eigenvalues and eigenvectors. 

A major strength of the book is the way it connects the algebraic and geometric sides of the subject. Roman develops real and complex inner-product spaces and the structure theory of normal operators, culminating in finite-dimensional spectral theory. From there, the scope becomes unusually wide: bilinear forms, metric spaces, Hilbert spaces, tensor products, convexity and separation, affine geometry, singular values, and the Moore–Penrose inverse all receive attention. The third edition also contains a chapter on associative algebras, including Frobenius's characterization of finite-dimensional real division algebras and Wedderburn's theorem for finite division algebras, as well as material on the spectral mapping theorem.

This breadth makes Advanced Linear Algebra closer to a comprehensive reference work than a conventional linear-algebra textbook. Particularly valuable is Roman's treatment of topics that students often encounter separately in abstract algebra, functional analysis, geometry, or numerical linear algebra. Tensor products, Hilbert spaces, canonical forms, normal operators and associative algebras are presented as parts of a larger mathematical structure rather than as unrelated techniques. The final treatment of the umbral calculus is especially unusual for a linear-algebra textbook and illustrates just how far Roman is willing to extend the subject. 

The price of this breadth is difficulty. Although Roman develops the necessary basic linear algebra, the presentation is rapid, and Springer describes prior linear algebra plus a degree of mathematical maturity as highly desirable.  This is therefore not the ideal first encounter with linear algebra. It is much better suited to someone who already understands matrices, vector spaces, bases, linear transformations, eigenvalues and elementary proofs and now wants to understand why the subject has the structure it does. The exercises and proof-oriented presentation also make it particularly appropriate for graduate study and preparation for more advanced work in algebra, geometry and analysis.

Key Takeaways

• Structural rather than computational: the emphasis is on vector spaces, transformations, modules, operators and proofs rather than routine matrix calculations.
  
• Exceptionally broad: it progresses from classical linear algebra to modules, spectral theory, Hilbert spaces, tensor products, convexity, affine geometry, generalized inverses and associative algebras. 
  
• Best for experienced readers: some previous linear algebra and mathematical maturity are strongly recommended.
  
• Excellent reference: its breadth makes it useful well beyond a single graduate course; reviewers cited by Springer specifically praise its comprehensive coverage, proofs and exercises. 
  
  

Overall:Advanced Linear Algebra is an excellent choice for a reader who wants to make the transition from elementary linear algebra to the more abstract viewpoint used in modern mathematics. It is demanding, but its combination of depth, breadth and structural insight makes it particularly valuable as both a graduate textbook and a long-term mathematical reference.


BOOK]]> https://mklab.gr/showthread.php?tid=1649 Mon, 17 Aug 2026 18:43:56 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1649 Author: Gilbert Strang
Publisher: Wellesley-Cambridge Press
First published: 1993

Gilbert Strang’s Introduction to Linear Algebra is one of the best-known undergraduate introductions to the subject, built around understanding what matrices and vectors actually do rather than treating linear algebra as a collection of mechanical procedures. The book develops the subject from vectors and linear combinations through solving systems $Ax=b$, vector spaces and the four fundamental subspaces, orthogonality and least squares, determinants, eigenvalues and eigenvectors, and singular values. Matrix factorizations such as $A=LU$, $A=QR$, and the singular value decomposition play a central role, connecting seemingly separate ideas into a coherent picture. 

A major strength of Strang's approach is its balance between geometric intuition, computation, and applications. Rather than beginning with a highly abstract axiomatic treatment, Strang repeatedly interprets algebraic operations geometrically—column spaces, nullspaces, projections, orthogonality, rank, and eigenvectors become objects the reader can visualize and reason about. The newer sixth edition introduces independent columns, rank, and column space earlier and extends the applications to modern areas such as optimization and learning from data. This makes the text particularly relevant to students interested not only in pure mathematics but also numerical analysis, statistics, data science, machine learning, engineering, and scientific computing. 

The book is also unusually well suited to self-study because Strang's informal explanatory style is complemented by examples, exercises, computational material, and his famous MIT linear algebra lectures. The companion resources provide sample sections and additional learning materials. For someone who wants to understand linear algebra rather than simply learn algorithms for manipulating matrices, this remains an excellent choice: conceptually serious without being unnecessarily formal, and particularly strong at revealing the relationships between equations, geometry, matrix factorizations, and applications.

Key takeaways

• Linear algebra is about structure, not just matrix calculation: concepts such as column space, nullspace, rank, and linear independence explain why systems behave as they do.
  
• Matrix factorizations unify the subject: $LU$, $QR$, eigenvalue decompositions, and SVD provide different ways of exposing the structure of a matrix.
  
• Geometry is fundamental: projections, orthogonality, subspaces, and eigendirections give geometric meaning to algebraic calculations.
  
• Highly relevant to modern mathematics and computing: the material provides much of the mathematical foundation for optimization, statistics, numerical computation, data science, and machine learning. 
  

Recommended for: undergraduate mathematics, engineering, computer science, physics, statistics, and data-science students, as well as independent learners who prefer intuition and applications alongside mathematical theory.


Gilbert Strang's official Introduction to Linear Algebra resources]]> Author: Gilbert Strang
Publisher: Wellesley-Cambridge Press
First published: 1993

Gilbert Strang’s Introduction to Linear Algebra is one of the best-known undergraduate introductions to the subject, built around understanding what matrices and vectors actually do rather than treating linear algebra as a collection of mechanical procedures. The book develops the subject from vectors and linear combinations through solving systems $Ax=b$, vector spaces and the four fundamental subspaces, orthogonality and least squares, determinants, eigenvalues and eigenvectors, and singular values. Matrix factorizations such as $A=LU$, $A=QR$, and the singular value decomposition play a central role, connecting seemingly separate ideas into a coherent picture. 

A major strength of Strang's approach is its balance between geometric intuition, computation, and applications. Rather than beginning with a highly abstract axiomatic treatment, Strang repeatedly interprets algebraic operations geometrically—column spaces, nullspaces, projections, orthogonality, rank, and eigenvectors become objects the reader can visualize and reason about. The newer sixth edition introduces independent columns, rank, and column space earlier and extends the applications to modern areas such as optimization and learning from data. This makes the text particularly relevant to students interested not only in pure mathematics but also numerical analysis, statistics, data science, machine learning, engineering, and scientific computing. 

The book is also unusually well suited to self-study because Strang's informal explanatory style is complemented by examples, exercises, computational material, and his famous MIT linear algebra lectures. The companion resources provide sample sections and additional learning materials. For someone who wants to understand linear algebra rather than simply learn algorithms for manipulating matrices, this remains an excellent choice: conceptually serious without being unnecessarily formal, and particularly strong at revealing the relationships between equations, geometry, matrix factorizations, and applications.

Key takeaways

• Linear algebra is about structure, not just matrix calculation: concepts such as column space, nullspace, rank, and linear independence explain why systems behave as they do.
  
• Matrix factorizations unify the subject: $LU$, $QR$, eigenvalue decompositions, and SVD provide different ways of exposing the structure of a matrix.
  
• Geometry is fundamental: projections, orthogonality, subspaces, and eigendirections give geometric meaning to algebraic calculations.
  
• Highly relevant to modern mathematics and computing: the material provides much of the mathematical foundation for optimization, statistics, numerical computation, data science, and machine learning. 
  

Recommended for: undergraduate mathematics, engineering, computer science, physics, statistics, and data-science students, as well as independent learners who prefer intuition and applications alongside mathematical theory.


Gilbert Strang's official Introduction to Linear Algebra resources]]>