Linear Algebra

Studying Linear Algebra

Row reduction answers computational questions, while vector spaces and linear maps explain why the answers fit together. Hefferon keeps elimination and proof in view; Axler places more weight on operators, invariant subspaces and spectral structure. If you can find a null space but cannot explain rank-nullity, the operator route can expose the gap. If changes of basis still feel mysterious, keep concrete matrices beside that route.

What to work on

  • Compute kernels and images and prove rank-nullity.
  • Express the same map in two bases and derive the change-of-basis formula.
  • Decide diagonalizability using eigenspaces or the minimal polynomial.
  • State the real and complex spectral theorems with the correct hypotheses.

A starting text

Linear Algebra Done Right, fourth edition — Sheldon Axler. Read 3B, Null Spaces and Ranges, and the Change of Basis subsection of 3D. Then use 5B and 5D for the minimal polynomial and diagonalizability; finish with 7A-7B for adjoints and the spectral theorem after reviewing inner products in chapter 6 if needed.

Choosing what to read

Express one map in two bases and derive the coordinate conversion instead of memorizing the formula. Work with differentiation on a finite-dimensional polynomial space, then compare a Jordan block with a real symmetric matrix. State the field and inner product before invoking a spectral theorem. After a calculation, write one sentence explaining which feature belongs to the map itself and which feature depends on your coordinates.

Linear Algebra Done Right, fourth edition

Sheldon Axler

Explain kernels, minimal polynomials and spectral decompositions conceptually.

How it reads
Operator-centered treatment of finite-dimensional structure.
What it expects
Proof-writing; bases and dimension, with chapter 2 available for review.
What to keep in mind
An operator emphasis does not replace practice solving systems or tracking coordinates.
Practice
Exercises provided; public solutions not verified. Exercises 3B, 3D, 5B, 5D and 7B; each set follows its named section.

Complete public PDF.

Edition, reading range and source

Read 3B, Null Spaces and Ranges, and the Change of Basis subsection of 3D. Then use 5B and 5D for the minimal polynomial and diagonalizability; finish with 7A-7B for adjoints and the spectral theorem after reviewing inner products in chapter 6 if needed.

Source · Checked 2026-10-16

Linear Algebra

Jim Hefferon

Connect row operations to subspaces before an operator-focused second course.

How it reads
Elimination and vector spaces developed together with proofs.
What it expects
The catalogue lists one semester of calculus.
What to keep in mind
Keep the abstract map visible rather than treating every argument as an elimination algorithm.
Practice
The author page states that the solutions book gives a worked answer to every exercise; exercise-to-solution coverage was not independently audited.

Author page provides a free book download route and a separate solutions-book link.

Edition, reading range and source

Use the systems, vector-space and linear-map portions of the author edition; confirm its current contents before assigning pages.

Source · Checked 2026-10-17

Practice

Linear-map and spectral exercises. Exercises 3B, 3D, 5B, 5D and 7B; each set follows its named section.. Exercises provided; public solutions not verified.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

  1. For D on the real polynomials of degree at most 3, where Dp = p', compute kernel, image and a matrix, and verify rank-nullity.

    A hint

    Apply DD to the basis (1,x,x2,x3)(1,x,x^2,x^3). The coordinate vectors of the results form the columns of the matrix.

    Work through the solution

    Take both domain and codomain to be P3(R)P_3(\mathbb R) and use the ordered basis (1,x,x2,x3)(1,x,x^2,x^3). Since D(1)=0D(1)=0, D(x)=1D(x)=1, D(x2)=2xD(x^2)=2x and D(x3)=3x2D(x^3)=3x^2,

    [D]=(0100002000030000).[D]=\begin{pmatrix}0&1&0&0\\0&0&2&0\\0&0&0&3\\0&0&0&0\end{pmatrix}.

    A polynomial has zero derivative exactly when it is constant, so ker⁡D=span⁡(1)\ker D=\operatorname{span}(1). Every polynomial of degree at most two has an antiderivative of degree at most three, so im⁡D=P2(R)=span⁡(1,x,x2)\operatorname{im}D=P_2(\mathbb R)=\operatorname{span}(1,x,x^2). Thus nullity⁡D=1\operatorname{nullity}D=1 and rank⁡D=3\operatorname{rank}D=3, with 1+3=4=dim⁡P3(R)1+3=4=\dim P_3(\mathbb R).

  2. Compare the matrices [[1,1],[0,1]] and [[2,1],[1,2]]: find eigenspaces, determine diagonalizability, and give an orthonormal eigenbasis when one exists.

    A hint

    A repeated root of the characteristic polynomial does not by itself decide diagonalizability. Count independent eigenvectors. For the symmetric matrix, test (1,1)(1,1) and (1,−1)(1,-1).

    Work through the solution

    For A=(1101)A=\begin{pmatrix}1&1\\0&1\end{pmatrix}, the only eigenvalue is 11. The equation (A−I)(u,v)=0(A-I)(u,v)=0 forces v=0v=0, so its eigenspace is span⁡((1,0))\operatorname{span}((1,0)). Its dimension is only one; AA is not diagonalizable.

    For B=(2112)B=\begin{pmatrix}2&1\\1&2\end{pmatrix}, the eigenspaces are E3=span⁡((1,1))E_3=\operatorname{span}((1,1)) and E1=span⁡((1,−1))E_1=\operatorname{span}((1,-1)). The vectors

    u=12(1,1),v=12(1,−1)u=\frac{1}{\sqrt2}(1,1),\qquad v=\frac{1}{\sqrt2}(1,-1)

    form an orthonormal eigenbasis. With QQ having columns u,vu,v, one has QTBQ=diag⁡(3,1)Q^TBQ=\operatorname{diag}(3,1). Symmetry explains why an orthonormal choice is available here; diagonalizability alone would not guarantee it.

Already comfortable with this? Skip the map review if you can explain rank-nullity without row-reduction alone and derive change of basis. Skip the spectral selection only if you can distinguish diagonalizable, normal and self-adjoint operators over the appropriate field.

Review the prerequisites · Compare the full subject collection

Where this can lead

Reading-source dates
  • Linear Algebra Done Right, fourth edition · 2026-10-16

Reading choices revised 2026-10-17.

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Linear Algebra

Online text

Jim Hefferon · Textbook · Introductory

Balances Gaussian elimination with vector spaces and proofs. Worked solutions to all exercises are useful for independent study; suitable before an operator-focused second course.

Preparation: One semester of calculus.

Full book & solutionsReviews / assessment
Further print reading (3 retained references)

From the original catalogue. These are bibliographic references, not free full-text links; an authorized complete free edition is not currently linked here.

  • Linear Algebra — Kenneth Hoffman and Ray Kunze
  • Linear Algebra, 4th edition — Stephen H. Friedberg, Arnold J. Insel and Lawrence E. Spence
  • Honors in Linear and Abstract Algebra I — Ivan Ip

Additional reading

Search the complete catalogue.

Introductory reading

Understanding Linear Algebra

David Austin · Textbook · Introductory · English · Full-text access unverified

Develops matrices and vector spaces through geometric ideas and applications. The author route provides online and PDF editions.

Official page / access details
Edition and sources

Checked 2026-10-11

Linear Algebra

David Cherney, Tom Denton, Rohit Thomas and Andrew Waldron; edited by Katrina Glaeser and Travis Scrimshaw · Textbook · Introductory · English · PDF

Complete 436-page UC Davis PDF opened; title page states CC BY-NC-SA 3.0.

Connects linear systems and vector-space reasoning with optimization, orthogonal bases and singular values.

Preparation: School algebra; elementary functions.

Use: Study computations together with conceptual and written review problems.

Open full PDFSource / terms
Edition and sources

Version: First edition, 2013; institutional download page lists update August 24, 2016 · Checked 2026-10-13

Linear Algebra

Lars Hesselholt and Nathalie Wahl · Textbook · Introductory · Danish · PDF

Full 281-page book opened on coauthor Hesselholt's university site; linked by the department archive.

English title: Linear Algebra. A first course that presents matrix methods and vector spaces with attention to proofs and the order of multiplication.

Preparation: School algebra; elementary set notation.

Use: Study computations together with definitions and proofs, initially using real or complex scalars.

Open full PDFSource / terms
Edition and sources

Original title: Lineær Algebra

Version: Second edition, October 2017, as printed in the inspected PDF · Checked 2026-10-13

Linear Algebra: Principles and Applications

李思 (Si Li) · Course notes · Introductory · Chinese · PDF

Public full text; checked against the title and contents.

Develops vector spaces, matrices, determinants, eigenvalues and inner products, followed by tensor constructions, differential equations and matrix groups. The author identifies Chapters 1–5 as the required core; later chapters are optional.

Preparation: School algebra and elementary functions.

Use: Use Chapters 1–5 for a first course; choose later applications after that foundation.

Open full PDFSource / terms
Edition and sources

Original title: 线性代数

Version: 2025-12 author PDF · Checked 2026-11-03

An Introduction to Determinants

Determinantulo · Course notes · Introductory · Chinese · PDF

Public full text; checked against the title and contents.

A book devoted to determinants, including their definitions, identities and methods of calculation. Use selected chapters alongside a linear-algebra course rather than replacing the whole course with determinant techniques.

Preparation: School algebra and basic matrix operations.

Use: Consult the introductory chapters for definitions, then select the identities needed for a particular problem.

Open full PDFSource / terms
Edition and sources

Original title: 行列式入门

Version: Version 2π, 2025-06-28 · Checked 2026-11-03

Intermediate reading

Linear Algebra Done Wrong

Sergei Treil · Textbook · Intermediate · English · Full-text access unverified

A proof-oriented text emphasizing linear maps, bases, inner products and spectral theory. Includes bilinear forms, duality and tensors.

Official page / access details
Edition and sources

Checked 2026-10-11

Finite-Dimensional Vector Spaces

Paul R. Halmos · Textbook · Intermediate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A structural linear-algebra course emphasizing vector spaces and linear maps. A concise bridge beyond row reduction into invariant reasoning.

Preparation: First linear algebra; proof-writing.

Official page / access details
Edition and sources

Version: Second edition, 1958; publisher reprint · Checked 2026-10-11

Linear Algebra — supplementary notes for Math 3000

Pete L. Clark · Course notes · Intermediate · English · PDF

Moves from matrix computations to vector spaces, linear maps, determinants, eigenvalues and duality.

Preparation: Basic matrix manipulation and mathematical proofs.

Use: Bridge computational linear algebra and abstract algebra.

Open full PDFSource / terms
Edition and sources

Version: Author-hosted notes; date not stated · Checked 2026-10-13

A Tutorial on Spectral Clustering

Ulrike von Luxburg · Survey · Intermediate · English · PDF

Public full text; checked against the title and contents.

Derives graph-Laplacian clustering algorithms and compares their normalized and unnormalized forms.

Preparation: Eigenvalues; graphs; elementary probability.

Use: Derive each graph Laplacian and compare the algorithm variants on the same graph.

Open full PDFSource / terms

Found via — References: A Tutorial on Spectral Clustering

Edition and sources

Version: arXiv 0711.0189; inspected electronic version · Checked 2026-11-03

Graduate reading

Advanced Linear Algebra

Steven Roman · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Develops structural linear algebra beyond matrix manipulation, including module-theoretic viewpoints and additional advanced topics.

Preparation: A proof-based linear-algebra course; rings and modules helpful.

Official page / access details
Edition and sources

Version: Second edition, 2005 · Checked 2026-10-11

Matrix Analysis

Roger A. Horn and Charles R. Johnson · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

An extensive second course covering canonical forms, matrix norms, positivity and spectral questions. Its many problems are useful for mastering matrix arguments.

Preparation: Proof-based linear algebra; some real/complex analysis.

Official page / access details
Edition and sources

Version: Second edition; publisher release 2012, imprint 2013 · Checked 2026-10-11

Matrix Analysis

Rajendra Bhatia · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Studies matrix inequalities and spectral structure from an analytic viewpoint. Distinct from the Horn–Johnson book with the same title.

Preparation: Linear algebra, norms and elementary analysis.

Official page / access details
Edition and sources

Version: 1997 · Checked 2026-10-11

Numerical Linear Algebra

Lloyd N. Trefethen and David Bau · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Develops stable matrix computation, decompositions and iterative methods. Its numerical emphasis complements structural algebra; it does not replace proof-based linear algebra.

Preparation: Linear algebra; norms; elementary numerical analysis.

Official page / access details
Edition and sources

Version: Twenty-fifth anniversary edition, 2022 · Checked 2026-10-11

Topics in Matrix Analysis

Roger A. Horn and Charles R. Johnson · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A separate companion to Matrix Analysis on numerical ranges, singular values, matrix equations and related topics. Chapters and reprints count as one work.

Preparation: Advanced linear algebra and the basic matrix-analysis toolkit.

Official page / access details
Edition and sources

Version: First published 1991; consult publisher for reprint date · Checked 2026-10-11

Spectral Clustering: Seminar Notes

Zeibniz · Course notes · Graduate · Chinese · Online text

Online lecture text; chapters can be read separately.

An applied seminar on graph-based clustering and its spectral interpretation, connecting matrix eigenvectors, graph Laplacians and statistical learning.

Preparation: Linear algebra; elementary graphs and probability.

Use: Read after eigenvalues and orthogonal diagonalization; compare the derivations with von Luxburg’s tutorial.

Open online text / archive
Edition and sources

Original title: 谱聚类

Version: Fall 2020 seminar · Checked 2026-11-03

Research reading

Randomized Numerical Linear Algebra: Foundations & Algorithms

Per-Gunnar Martinsson and Joel A. Tropp · Survey · Research · English · PDF

Full 116-page survey opened on coauthor Tropp's Caltech website.

A substantial survey connecting random embeddings and matrix sampling to least squares, low-rank approximation and linear solvers.

Preparation: Linear algebra, numerical matrix methods and elementary probability.

Use: Bridge a numerical linear algebra course to randomized algorithms and their analysis.

Open full PDFSource / terms
Edition and sources

Version: arXiv version 3, March 15, 2021 · Checked 2026-10-13

Chinese lecture notes

  • 线性代数 — 李思 (Si Li). Use Chapters 1–5 for a first course; choose later applications after that foundation.