Linear Algebra
Essence of Linear Algebra (18): Frontiers and Summary
Series finale: quantum gates as unitary matrices, graph convolution as Laplacian filtering, attention as soft retrieval, LoRA as low-rank adaptation, tensor networks, the matrix exponential, free probability, and a …
Essence of Linear Algebra (17): Linear Algebra in Computer Vision
Images are matrices, geometric transformations are matrix multiplications, camera imaging is a projective map, and 3D reconstruction is solving linear systems. Master the linear algebra that quietly powers every corner …
Essence of Linear Algebra (16): Linear Algebra in Deep Learning
Deep learning is large-scale matrix computation. From backpropagation as the chain rule in matrix form, to im2col turning convolutions into GEMM, to attention as soft retrieval via dot products -- see every core DL …
Essence of Linear Algebra (15): Linear Algebra in Machine Learning
Machine learning speaks linear algebra as its native language. From PCA to SVMs, from matrix factorization in recommender systems to gradient descent optimization -- see how vectors, matrices, and decompositions power …
Essence of Linear Algebra (14): Random Matrix Theory
Fill a huge matrix with random numbers, compute its eigenvalues, and watch stunning regularity emerge from chaos. Learn the Wigner semicircle law, Marchenko-Pastur distribution, and Tracy-Widom limit -- with applications …
Essence of Linear Algebra (13): Tensors and Multilinear Algebra
From scalars to high-dimensional data cubes -- tensors generalize vectors and matrices to arbitrary dimensions. Learn CP and Tucker decomposition, see how tensors compress neural networks and power recommender systems, …
Sparse Matrices and Compressed Sensing -- Less Is More
Sparsity is everywhere: JPEG photos, MRI scans, genomic data. Compressed sensing exploits this to recover signals from far fewer measurements than traditional theory requires. This chapter covers L1 regularization, …
Matrix Calculus and Optimization -- The Engine Behind Machine Learning
Adjusting the shower temperature is a tiny version of training a neural network: you change a parameter based on an error signal. Matrix calculus is the language that scales this idea to millions of parameters, and …
Matrix Norms and Condition Numbers -- Is Your Linear System Healthy?
The condition number is a 'health report' for a linear system -- it tells you whether tiny input errors will explode into catastrophic output errors. This chapter covers vector norms, matrix norms, the spectral norm, …
Singular Value Decomposition -- The Crown Jewel of Linear Algebra
SVD decomposes any matrix -- not just square or symmetric ones. From image compression to Netflix recommendations, from face recognition to gene analysis, SVD is the most powerful and most universal decomposition in …
Symmetric Matrices and Quadratic Forms -- The Best Matrices in Town
Symmetric matrices are the 'nicest' matrices in linear algebra: real eigenvalues, orthogonal eigenvectors, and perfect diagonalization. This chapter builds intuition for quadratic forms, positive definiteness, and why …
Orthogonality and Projections -- When Vectors Mind Their Own Business
Orthogonality is what makes GPS work, noise-canceling headphones cancel, and JPEG compress. This chapter builds geometric intuition for orthogonal vectors, projections, Gram-Schmidt, QR decomposition, and least squares …
Eigenvalues and Eigenvectors
Some special vectors survive a matrix transformation with their direction intact -- they only get scaled. These eigenvectors and their eigenvalues reveal the deepest structure of linear transformations, powering …
Linear Systems and Column Space
When does Ax = b have a solution? How many? The honest answer is geometric: it depends on whether b lives inside the column space of A, and on how much of the input space A crushes to zero. This chapter weaves Gaussian …
The Secrets of Determinants
Determinants are not just tedious calculations -- they measure how much a transformation stretches or compresses space. This chapter gives you the geometric intuition behind determinants, their key properties, and …
Matrices as Linear Transformations
Matrices are not tables of numbers -- they are machines that transform space. This chapter shows you how to see rotation, scaling, shearing, reflection, and projection as matrices, and why matrix multiplication means …
Linear Combinations and Vector Spaces
If vectors are building blocks, linear combinations are the blueprint. This chapter develops the five concepts that the rest of linear algebra is built on: span, linear independence, basis, dimension, and subspaces.
The Essence of Vectors -- More Than Just Arrows
Vectors are everywhere -- from GPS navigation to Netflix recommendations. This chapter builds your intuition from arrows in space to abstract vector spaces, covering addition, scalar multiplication, inner products, …