Applied Linear Algebra Labs

Hands-on Jupyter and Python notebooks that teach applied linear algebra through real projects: image compression with SVD, PCA on MNIST, Google's PageRank algorithm, spectral clustering, quantum computing basics, word embeddings, and more. Useful for anyone learning linear algebra by writing code.

Each lab ships in one or more versions (v1 through v7). Later versions cover the same core ideas from a different angle or add new topics, so you can work through several versions of the same lab without repeating yourself.

Table of contents

• Lab 1: NumPy, transformations, and ciphers
• Lab 2: Least squares, QR decomposition, and quantum computing
• Lab 3: SVD, PCA, PageRank, and spectral methods
• Lab 4: Word embeddings with SVD
• Running the notebooks

Lab 1: NumPy fundamentals, 3D graphics, and cryptography

Version	Topics	Notebook
v1	NumPy arrays, array arithmetic, slicing, Gaussian elimination, interpolation, color grading	Lab#01-Introduction.ipynb
v2	NumPy, color grading, linear transformations, 3D polygon rendering with orthogonal matrices	Lab#01-V2-Introduction.ipynb
v3	NumPy, Gaussian elimination, polynomial interpolation, Hill cipher encryption and decryption, known plaintext attack on the Hill cipher	Lab#-1-V3-Introduction.ipynb
v4	NumPy, linear systems, image processing with color channel operations, 2D and 3D geometric transformations (scaling, rotation, shearing, translation)	FUM-Linear-Algebra-lab#01.ipynb

Lab 2: Least squares, regression, and quantum computing

Version	Topics	Notebook
v1	Projection matrices and least squares, solving least squares via QR decomposition with modified Gram-Schmidt, medical cost prediction with linear regression	Lab#02-Least-Mean-Squares.ipynb
v2	Least squares and QR decomposition, car price prediction, polynomial regression	Lab#02-V2-Least-Mean-Squares.ipynb
v3	Quantum computing foundations: qubits, superposition and measurement probabilities, basis changes and Gram-Schmidt orthonormalization, quantum gates as matrices, BB84 quantum key distribution	FUM-Linear-Algebra-lab#02.ipynb

Lab 3: SVD, PCA, PageRank, and spectral methods

Version	Topics	Notebook
v1	Image compression with singular value decomposition (SVD), video background subtraction	Lab#03-SVD.ipynb
v2	SVD image compression, eigenfaces and face reconstruction on the Labeled Faces in the Wild dataset	Lab#03-V2-SVD-PCA.ipynb
v3	Encoding data as matrices, SVD, principal component analysis (PCA), dimensionality reduction on MNIST	Lab#03-v3-SVD-PCA.ipynb
v4	Determinants, Cramer's rule, eigenvectors via power iteration, and Google's PageRank algorithm for ranking web pages by link structure	FUM_Linear_Algebra_lab_#03.ipynb
v5	Spectral clustering: grouping data based on pairwise similarities and graph structure	FUM_Linear_Algebra_lab_#03-Spectral.ipynb
v6	Spectral data selection for identifying informative samples within training batches	FUM_Linear_Algebra_lab_#03.ipynb
v7	Leverage scores and SVD-based dimensionality reduction for anomaly detection in industrial images	FUM_Linear_Algebra_lab_#03.ipynb

Example: Background subtraction with SVD (from v1): original clip minus its low-rank approximation leaves the moving foreground.

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Lab 4: Natural language processing with SVD

Version	Topics	Notebook
v1	Implementing SVD, text data loading and preprocessing, vocabulary construction, word embeddings via singular value decomposition	linalg_proj4.ipynb

Running the notebooks

Install uv and set up the project and install the dependencies:

uv init
uv add numpy matplotlib pandas scikit-learn jupyterlab

Then clone and launch:

git clone https://github.com/arashsm79/applied-linear-algebra-lab.git
cd applied-linear-algebra-lab
uv run jupyter lab

Contributing

Contributions are welcome. Open an issue if you spot a mistake in a notebook, or send a pull request with a fix or a new lab topic.

License

Distributed under the MIT License.