Learn linear algebra as computation, geometry, and structure
Linear algebra studies vectors, matrices, systems of linear equations, and the transformations that connect them. This suite emphasizes the meaning behind the algorithms: what a calculation does, why it works, how to check it, and where it appears in science, engineering, computing, and data analysis.
The central ideas
See one connected subject rather than a collection of unrelated procedures
Represent relationships
A matrix encodes coefficients, data, or a linear transformation. The equation Ax=b unifies systems, inverse problems, models, and algorithms.
Transform space
Multiplication by a matrix moves, scales, rotates, reflects, shears, or projects vectors while preserving linear combinations.
Reveal structure
Orthogonality separates independent directions; eigenvectors expose invariant directions; singular values identify principal stretch and numerical sensitivity.
Course modules
Work sequentially or enter at the topic needed for your course
Matrix Fundamentals
Notation, dimensions, matrix types, transpose, addition, scalar multiplication, trace, and structural classification.
02Not completedMatrix Multiplication
Dimension rules, row-column dot products, column combinations, composition, and noncommutativity.
03Not completedLinear Systems and RREF
Augmented matrices, elimination, pivots, free variables, rank, consistency, and complete solution classification.
04Not completedDeterminants and Inverses
Signed area or volume, cofactor expansion, determinant properties, singularity, and matrix inversion.
05Not completedVectors and Vector Spaces
Linear combinations, span, subspaces, independence, bases, dimension, rank, and the four fundamental subspaces.
06Not completedOrthogonality and Projections
Inner products, angles, orthogonal complements, projections, Gram–Schmidt, QR, and residual geometry.
07Not completedLinear Transformations
Linearity tests, standard matrices, kernel and image, rank-nullity, composition, and change of basis.
08Not completedEigenvalues and Eigenvectors
Characteristic polynomials, eigenspaces, multiplicity, diagonalization, spectral structure, and power iteration.
09Not completedLeast Squares and SVD
Overdetermined systems, normal equations, residuals, line fitting, singular values, pseudoinverses, and PCA.
RNot completedCumulative Review Center
Randomized quizzes, flashcards, practice generators, a searchable glossary, and an exam-preparation checklist.
Suggested learning paths
Choose a sequence that matches the purpose of the course
Foundations path
- Matrix fundamentals
- Multiplication
- Linear systems
- Vector spaces
- Determinants
- Orthogonality
- Transformations
- Eigenanalysis
- Least squares and SVD
Geometry path
- Vectors and vector spaces
- Linear transformations
- Determinants
- Orthogonality
- Eigenvectors
- SVD geometry
Applied path
- Matrix operations
- Systems and rank
- Orthogonality
- Least squares
- Eigenanalysis
- SVD, conditioning, and PCA
Readiness diagnostic
Use these questions to identify concepts that deserve extra attention
Which description best captures a vector?
In the product AB, what must match?
What does a pivot column indicate in a matrix?
Which idea connects orthogonal projection and least squares?
Why are eigenvectors useful?
External learning resources
Verified references for lectures, visual intuition, exercises, and numerical computing
MIT OpenCourseWare 18.06SC
Gilbert Strang’s independent-study course includes lectures, notes, problem sets, problem-solving videos, and exams.
Open MIT course ↗3Blue1Brown: Linear Algebra
Geometric explanations of span, transformations, determinants, eigenvectors, dot products, and change of basis.
Open visual lessons ↗Interactive Linear Algebra
An open textbook by Dan Margalit and Joseph Rabinoff covering systems, transformations, determinants, eigenvalues, and orthogonality.
Open textbook ↗Khan Academy Linear Algebra
Short instructional sequences and practice covering vectors, matrices, transformations, bases, and alternate coordinate systems.
Open practice course ↗NumPy linear algebra reference
Official documentation for computational functions such as solve, det, matrix_rank, eig, svd, norm, and condition number.
Open NumPy documentation ↗GeoGebra Matrix Calculator
An additional browser tool for checking matrix arithmetic, determinants, inverses, and rank.
Open matrix calculator ↗Notation quick reference
Read symbols before beginning the first computational module
| Notation | Meaning | Example interpretation |
|---|---|---|
| A ∈ Rᵐˣⁿ | A is an m-row, n-column real matrix. | A maps vectors in Rⁿ into Rᵐ. |
| aᵢⱼ | Entry in row i, column j. | The first index selects the output row. |
| span{v₁,…,vₖ} | Every linear combination of the listed vectors. | All reachable vectors using scalar coefficients. |
| N(A) | Null space or kernel of A. | All x for which Ax=0. |
| C(A) | Column space of A. | All possible outputs Ax. |
| Aᵀ | Transpose of A. | Rows become columns. |
| λ, v | Eigenvalue and eigenvector. | Av=λv with v≠0. |
| σᵢ | Singular value. | Principal stretch magnitude. |
Recommended starting point
Begin with Matrix Fundamentals even when matrices are familiar. The terminology introduced there—shape, column, transpose, trace, rank, and structural type—is used throughout every subsequent page.