Mathematical functions, domain and range
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…
linear algebra
What is a vector space and examples
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…
Basis of vector space
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…
Connection Between Positive Definite Matrices and Bilinear Forms
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…
Understanding Set Identities Through Party Analogies
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…
Key Roles of Transformation Matrices in Regression and PCA
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…
Change of Basis in Vector Space
Change of basis in vector spaces is crucial in computer graphics and machine learning. This post provides explanations and concrete example on this matter.
Subspace and Examples
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.
Understanding Vector Spaces with Concrete Examples
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.
Matrix Properties and Remembering Tricks
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…
Tricks for remembering elementary matrix operations
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…