Linear Algebra II: Transformations, Eigenvalues & Orthogonality
A second linear algebra course on linear transformations, eigenvalues/eigenvectors and inner-product geometry, aligned with a typical Linear Algebra II syllabus.
What you’ll learn
- View matrices as linear transformations between vector spaces
- Compute and interpret eigenvalues and eigenvectors
- Decide when a matrix is diagonalizable and what that buys you
- Use inner products, orthogonality and projections in applications
Curriculum
M1
Linear transformations & matrix representations — 4 lessons
Show lessons
-
Linear transformations between vector spaces16m
-
Matrix representation of a linear map16m
-
Kernel, image, rank & nullity via transformations16m
-
Change of basis & similarity of matrices16m
M2
Eigenvalues, eigenvectors & diagonalization — 4 lessons
Show lessons
-
Eigenvalues & eigenvectors: definitions & intuition16m
-
Diagonalization & when it fails16m
-
Algebraic vs geometric multiplicity14m
-
Dynamics & repeated application of a matrix14m
M3
Inner products, orthogonality & least squares — 4 lessons
Show lessons
-
Inner products, norms & angles16m
-
Orthogonality & orthonormal bases16m
-
Gram–Schmidt & QR factorization16m
-
Orthogonal projections & least squares16m
Projects
- Analyse simple dynamical systems via eigenvalues/eigenvectors
- Implement Gram–Schmidt and QR factorisation in Python
- Solve least-squares problems and fit simple models to data
Prerequisites
- All core ideas from Linear Algebra I (vectors, matrices, rank, determinants)
- Comfort with basic proofs and algebraic manipulation
- Optional: some experience with Python/NumPy for experiments
Who is this for?
- Students in a second Linear Algebra course (eigenvalues & orthogonality)
- STEM majors heading towards differential equations, quantum, or ML
- Analysts who want to understand the linear algebra behind models
Outcomes
- Comfortable working with linear maps, eigendecompositions and orthogonality
- Able to solve and interpret least-squares and projection problems
- Ready for spectral methods, SVD and numerical techniques in Linear Algebra III
Resources
- Starter project / template (ZIP)
- Setup & study checklists (PDF)
- Core formulas / syntax cheatsheet (PDF)
Tip: Right-click → “Save link as…” if your browser opens the file.
FAQ
Is this beginner-friendly?
Yes. We start from mental models and build up with guided practice and plenty of worked examples.
Will I need extra software?
We stick to free or standard tools. Any optional extras are clearly marked and have alternatives.
Do I get updates?
Yes—lifetime access with updates as the field, tools, and exam expectations evolve.