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Leon 8th Edition · Complete Notes

Linear Algebra

All 7 chapters of the Leon textbook — definitions, proofs, worked examples, and interactive visualizations. Chapters 1–6 cover the standard undergraduate syllabus. Chapter 7 and select advanced sections go further into numerical methods, orthogonal polynomials, and Markov chains.

7 Chapters
Complete

These notes follow the Leon 8th Edition chapter-by-chapter. Chapters 1–6 cover the core undergraduate syllabus. Sections marked Beyond Syllabus — including orthogonal polynomials, nonneg matrices, Markov chains, and all of Chapter 7 — go further into the mathematics used in scientific computing, data science, and engineering.

Unit 1 — Matrices & Systems of Equations
01
Chapter 1
Matrices & Systems
of Equations
The fundamental problem of linear algebra: solving $A\mathbf{x}=\mathbf{b}$. Row reduction, echelon forms, and the complete theory of linear systems.
Linear Systems Row Echelon Form Gaussian Elimination Matrix Arithmetic Inverses Elementary Matrices LU Factorization Partitioned Matrices
6 sections 3 animations
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02
Chapter 2
Determinants
A single scalar encoding whether a matrix is invertible, how it scales volumes, and unlocking the characteristic equation for eigenvalues.
Cofactor Expansion Triangular Rule Row Ops & det Product Rule Row Reduction Method Cramer's Rule
2 sections 3 animations
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Unit 2 — Vector Spaces & Linear Transformations
03
Chapter 3
Vector Spaces
The abstract framework unifying $\mathbb{R}^n$, matrices, polynomials, and functions under 8 axioms. Subspaces, independence, bases, and dimension.
Axioms & Examples Subspaces Null & Column Space Linear Independence Basis & Dimension Rank-Nullity Theorem Change of Basis Row & Column Space
6 sections 2 animations
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04
Chapter 4
Linear Transformations
Structure-preserving maps between vector spaces. Every linear transformation has a matrix — and every matrix represents a linear transformation.
Definition & Examples Kernel & Image Gallery of Maps Matrix Representations Theorem 4.2.1 Similarity
3 sections 2 animations
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// Fundamental Results — Chapters 1–4
Matrix-vector product
$A\mathbf{x} = x_1\mathbf{a}_1 + \cdots + x_n\mathbf{a}_n$
Rank-Nullity Theorem
$\text{rank}(A)+\text{nullity}(A)=n$
Product Rule for det
$\det(AB)=\det(A)\cdot\det(B)$
Matrix of $L$ via basis
$[L(\mathbf{v})]_F = A[\mathbf{v}]_E$
Similarity
$B = S^{-1}AS$
Invertibility condition
$A^{-1}$ exists $\Leftrightarrow$ $\det(A)\neq0$
Unit 3 — Orthogonality
05
Chapter 5
Orthogonality
Projections, least squares fitting, inner product spaces, and the Gram-Schmidt process. Section 5.7 introduces orthogonal polynomials — the gateway to Fourier analysis and numerical integration.
Scalar Product Cauchy-Schwarz Projections Orthogonal Subspaces Least Squares Inner Product Spaces Gram-Schmidt / QR Orthogonal Polynomials +
7 sections 3 animations
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Unit 4 — Eigenvalues
06
Chapter 6
Eigenvalues
The intrinsic geometry of matrices. Eigenvalues underlie Google PageRank, quantum mechanics, PCA, and Markov chains. Includes SVD, quadratic forms, and the Perron-Frobenius theorem.
Characteristic Polynomial Diagonalization Hermitian Matrices SVD Quadratic Forms Positive Definite Nonneg Matrices + Markov Chains +
7 sections 3 animations
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Beyond the Course — Numerical Methods
07
Chapter 7
Numerical Linear Algebra Beyond Syllabus
What happens when you run linear algebra on a computer? Floating-point arithmetic, numerical stability, condition numbers, and the algorithms inside MATLAB, NumPy, and LAPACK.
Floating-Point & $\varepsilon_{\text{mach}}$ LU Factorization Partial Pivoting Matrix Norms Condition Numbers Householder Reflections Power Method QR Iteration Pseudoinverse
7 sections 5 animations
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What you'll need
  • High school algebra and some calculus
  • No prior linear algebra assumed
  • Each chapter builds on the previous ones
Textbook

Linear Algebra with Applications
Steven J. Leon — Pearson, 8th Edition.
Chapters and section numbers match the book exactly.

Advanced sections

Sections marked Beyond Syllabus go past the standard undergraduate course — connecting the theory to real applications in physics, economics, and numerical computing.

// Fundamental Results — Chapters 5–7
Normal equations
$A^TA\hat{\mathbf{x}} = A^T\mathbf{b}$
Spectral theorem
$A = U\Lambda U^T$ ($A$ symmetric)
SVD
$A = U\Sigma V^T$
Characteristic equation
$\det(A - \lambda I) = 0$
Diagonalization
$A = XDX^{-1}$, $\;A^k = XD^kX^{-1}$
Condition number
$\kappa(A) = \|A\|\|A^{-1}\| = \sigma_1/\sigma_n$
Linear Algebra · Complete Notes Steven J. Leon, Linear Algebra with Applications, Pearson 8th Ed.
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