Linear Algebra #

Introduction #

Linear algebra is a branch of mathematics that studies linear sets of equations and their transformation properties. It revolves around the concept of vectors, which are elements of a vector space. A vector space (or linear space) is a collection of objects called vectors, which can be added together and multiplied by scalars (real or complex numbers). Key concepts in linear algebra include matrices, determinants, and eigenvalues. Matrices represent linear transformations, which are functions between vector spaces that preserve vector addition and scalar multiplication. Determinants provide important information about a matrix, such as whether it is invertible (i.e., there exists a matrix that undoes the transformation represented by the original matrix).

An eigenvector of a square matrix is a non-zero vector that, when the matrix is multiplied by it, yields a scalar multiple of itself. This scalar is known as the eigenvalue associated with the eigenvector. Eigenvalues and eigenvectors have numerous applications, including in systems of differential equations, physics, engineering, and computer science. For instance, in Google’s PageRank algorithm, which is used to rank web pages in their search engine results, the web pages’ ranks are the components of an eigenvector of a matrix whose elements represent links between pages. Linear algebra is also essential in machine learning and data science, where it is used to handle and manipulate high-dimensional data.

Articles #

Linear Algebra Basics: Linear Algebra Basics covers the fundamentals of vectors, vector operations, matrices, and matrix operations
Cross Product: The article discusses the cross product of vectors in three-dimensional space, its geometric interpretation, important properties like anticommutativity and distributivity, its relation to other vector operations, examples of cross product calculations, and practice problems involving cross product applications.
Orthogonal Matrices: Square matrices whose columns and rows form orthonormal bases, their properties like being invertible with the inverse equal to the transpose, preserving vector norms, having determinants of ±1, eigenvalues of absolute value 1, and their applications in simplifying complex systems.
Wedge Product: The article discusses the wedge product of vectors in linear algebra, its geometric interpretation, important properties like anticommutativity and distributivity, its relation to other vector operations, examples of wedge product calculations, and practice problems involving wedge product applications.
WTF is Machine Learning?: An introduction to machine learning and walkthrough of building a simple neural network using PyTorch to classify handwritten digits from the MNIST dataset, covering data loading, defining the network architecture, training, evaluation, and making predictions.
The Frobenius Norm: The Frobenius norm of a matrix is defined as the square root of the sum of the absolute squares of its elements, providing a measure of the magnitude of the matrix.
Singular Value Decomposition: Singular Value Decomposition (SVD) is a factorization method in linear algebra that decomposes a matrix into a rotation, a rescaling, and another rotation, generalizing the eigendecomposition of a square normal matrix to any matrix.
Kolmogorov-Arnold Networks: Kolmogorov-Arnold Networks (KANs) are a novel neural network architecture that, inspired by the Kolmogorov-Arnold representation theorem, replaces the fixed activation functions of Multi-Layer Perceptrons (MLPs) with learnable activation functions on edges, leading to improved accuracy and interpretability.
Kolmogorov-Arnold Representation Theorem: The Kolmogorov-Arnold Representation Theorem states that every multivariate continuous function can be represented as a superposition of continuous functions of one variable and the binary operation of addition.
Time-series Forecasting: Time-series forecasting is a technique that uses historical and current data to predict future values, guiding strategic decision-making and understanding future trends.