A vector space (or linear space) is a fundamental concept in mathematics, specifically in linear algebra. It is a set of objects called vectors, along with two operations: vector addition and scalar multiplication, that satisfy… What is a vector space and examples
1. Basis of a Vector Space A basis of a vector space is a set of vectors that satisfy two key properties: Intuition: Properties: 2. Dimension of a Vector Space The dimension of a vector… Basis of vector space
A positive definite matrix and its connection to a bilinear form is a fundamental concept in linear algebra and geometry. Here’s a detailed explanation: Positive Definite Matrix A symmetric matrix of size is called positive… Connection Between Positive Definite Matrices and Bilinear Forms
Some common set identities explained with intuitive and fun analogies — think of sets like groups of people at a party: 1. Identity Laws 2. Domination Laws 3. Idempotent Laws 4. Complement Laws 5. Double… Understanding Set Identities Through Party Analogies
Statistical Context: Projection and transformation matrices appear frequently in statistics, especially in regression and PCA, where they play a crucial role in simplifying complex datasets and revealing underlying patterns. These matrices help in miniminimizemizing errors… Key Roles of Transformation Matrices in Regression and PCA
Change of basis in vector spaces is crucial in computer graphics and machine learning. This post provides explanations and concrete example on this matter.
The subspace is a subset of the vector space, preserving the operations and properties. This article presents the definition and examples, as well as counterexamples of this concept.
The vector space is an algebraic structure with properties of addition and multiplication, applied in many fields such as physics and machine learning. We will prove that Euclidean space and function space are vector spaces.
1. Basic Additive Properties of Matrices: Ways to remember: We see that the basic properties of matrix addition are quite similar to those of real numbers, with the commutative property, associative property, and the existence… Matrix Properties and Remembering Tricks
Connecting matrices to systems of linear equations can indeed help in better understanding and remembering the properties of matrices. By visualizing how matrix operations correspond to operations on systems of equations, abstract matrix properties become… Tricks for remembering elementary matrix operations
A function in mathematics and computer science is a relation between a set of inputs and a set of permissible outputs. It assigns each input exactly one output. Functions can be simple or complex, depending… Mathematical functions