Linear Algebra: Understanding Vectors, Matrices, and Transformations
Learn how vectors, matrices, systems of equations, and transformations work, with visual explanations and worked examples that build one idea at a time. Start with familiar algebra, then connect the ideas through projections, least squares, eigenvectors, and an introduction to data compression and PCA. Build a mathematical foundation used in physics, engineering, computer graphics, data science, machine learning, and AI.
See What a Matrix Is Doing
A matrix can describe a set of measurements, move a shape, or combine inputs into a prediction. The same calculation connects these jobs. This course helps you see that connection, then use it to solve problems and explain what your answer means.
We begin with arrows, coordinates, and familiar equations. Each new idea grows from a small example before the general notation appears. You will work with diagrams, adjustable examples, worked calculations, and questions with revealable solutions. Calculus is not required.
What You Will Learn
| Build the idea | Use it |
|---|---|
| Vectors and combinations | Describe movement, span, independence, and coordinates. |
| Matrices and systems | Apply linear maps, solve equations, and interpret all solution types. |
| Bases and geometry | Understand rank, inverse maps, determinants, and coordinate changes. |
| Orthogonality and fitting | Project vectors and find least-squares solutions using QR. |
| Eigenvectors and singular values | Study repeated transformations, symmetric matrices, approximation, and PCA. |
Where to Start
A working familiarity with negative numbers, fractions, equations, and coordinate graphs is enough to begin. The early lessons refresh the algebra we need. You can take this course before or after calculus. If a step feels unfamiliar, work the short example before opening its solution; the explanation is there to help you locate the missing idea.
Skills You’ll Develop
The broader competencies and practical skills this guide is designed to build.
Mathematical Relationships & Change
- Algebraic Reasoning
- Functions & Mathematical Modeling
Space & Measurement
- Geometric & Spatial Reasoning
Data & Uncertainty
- Statistical Reasoning & Data Interpretation
Guide Sections
Discover how a few numbers can change a shape—and open the door to a different way of thinking about math.
Practice substitution, exact arithmetic, equations, and ordered lists before using matrices.
Connect arrows, components, position, and higher-dimensional data.
Add vectors, scale them, and interpret linear combinations geometrically.
Use span to distinguish a line, a plane, and targets that cannot be formed.
Test linear independence and connect it with uniqueness of representations.
Use vector lengths and dot products while keeping measurement scales in view.
Identify matrix shape and use rows as weighted measurements.
Connect row calculations with span, reachability, and the images of coordinate directions.
Test preservation of combinations and distinguish linear maps from shifted or curved rules.
Build transformation matrices from the images of coordinate directions.
Calculate products by columns and explain why order matters.
Connect equations, column combinations, and geometric solution types.
Use legal row operations, an augmented matrix, and back substitution.
Interpret echelon and reduced forms without confusing coefficient and augmented columns.
Find column and null spaces and express all solutions as one solution plus a null direction.
Combine span and independence to give unique coordinates and explain rank-nullity.
Use subspace tests and coefficient coordinates for functions and other mathematical objects.
Connect reversibility with pivots and compute an inverse through row operations.
Calculate a 2 × 2 determinant and connect area, orientation, and invertibility.
Compute 3 × 3 determinants and connect them with the cross product.
Convert between bases and represent a transformation in a new coordinate system.
Decompose a vector into an along-line part and a perpendicular residual.
Use Gram–Schmidt, orthogonal coordinates, and the geometry of null spaces.
Build a line-fitting model, derive normal equations, and interpret residuals.
Factor independent columns and solve least-squares problems without forming AᵀA.
Calculate eigenvalues and eigenvectors and distinguish zero and repeated eigenvalues.
Use an eigenvector basis to calculate powers and recognize when diagonalization fails.
Analyze repeated updates and a two-state transition model.
Apply the spectral theorem and interpret positive and indefinite quadratic forms.
Build an SVD, interpret rank and null directions, and introduce the pseudoinverse.
Center data, identify principal directions, and quantify the information a rank reduction discards.
Distinguish residuals from sensitivity, use pivoting, and interpret condition numbers.
Build a system, diagnose inconsistency, fit it, and explain the residual geometrically.
Connect vectors, systems, transformations, and approximation, then choose what to practice next.
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