Applications of Vector Spaces
When an object is made up of multiple components it is often useful to represent the object as a vector, with one entry per component. The examples discussed in this section involve molecules, which are made up of atoms, and text documents, which are made up of words.
In some cases equations involving the objects give rise to vector equations. In other examples there are reasons to perform operations on the vectors using matrix algebra. Vector Spaces is also used in Machine Learning and its various other uses are:
- Data Representation: In many machine learning algorithms, data is represented as vectors. For example, images can be represented as vectors of pixel values, text documents can be represented as vectors of word counts or embeddings, and numerical data can be directly represented as vectors.
- Feature Vectors: Feature engineering involves creating meaningful representations of data. These representations are often in the form of feature vectors, where each feature corresponds to a dimension in the vector space. Feature vectors are used as input to machine learning models.
- Vector Operations: Vector operations such as addition, subtraction, dot products, and vector norms are commonly used in machine learning algorithms. For example, in clustering algorithms like k-means, vector addition and subtraction are used to calculate centroids.
- Linear Algebra in Models: Many machine learning models are based on linear algebra operations. For example, linear regression involves finding a line that best fits a set of data points, which can be formulated as a linear algebra problem involving vectors and matrices.
Vector Space- Definition, Axioms, Properties and Examples
A vector space is a group of objects called vectors, added collectively and multiplied by numbers, called scalars. Scalars are usually considered to be real numbers. But there are few cases of scalar multiplication by rational numbers, complex numbers, etc. with vector spaces.
In this article, we have covered Vector Space Definition, Axions, Properties and others in detail.
Table of Content
- What is Vector Space?
- Vector Space Axioms
- Vector Space Examples
- Dimension of a Vector Space
- Vector Addition and Scalar Multiplication
- Vector Space Properties
- Subspaces
Contact Us