Basis and Dimension in Vector Space

Vector Spaces are a fundamental concept in machine learning and mathematics, providing the structure for various operations in high-dimensional data processing. A vector space is a collection of objects called vectors that can be added together and multiplied by scalars (numbers) to produce new vectors, all while satisfying certain rules, such as closure under addition and scalar multiplication, the existence of a zero vector, and others.

Within vector spaces, the concepts of Basis and Dimension help define these spaces' underlying structure and capacity.

Basis of Vector Space

A basis is a set of vectors that serves as the "building blocks" of a vector space, allowing you to create any other vector in that space by scaling and combining these vectors.

For example, in a 2D plane (like a graph), the vectors [1, 0] (pointing right) and [0, 1] (pointing up) form a basis. With just these two directions, you can reach any point on the plane by moving a certain distance right and up.

This means a basis provides all the directions needed to describe any point in the space.

A basis has two essential properties:

• Linear Independence: No vector in the basis can be formed by combining the others.
• Spans the Space: Any vector in the space can be represented by scaling and adding these basis vectors.

Mathematical Definition

A subset S of a vector space V(F) is called a basis of V(F) if it satisfies two conditions:

• The set S is linearly independent.
• The set S generates V, which means every vector in V can be written as a linear combination of the vectors in S. In other words, V = ⟨S⟩, where S is the span of V.

Note: The span of a set of vectors is all possible vectors you can make by combining them in different ways.

Standard Basis

• For a 2-dimensional vector space V₂(F):
  • Standard Basis = {(1, 0), (0, 1)}

• For a 3-dimensional vector space V₃(F):
  • Standard Basis = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}

• For an n-dimensional vector space V_n​(F):
  • Standard Basis = {(1, 0, 0,. . . , 0), (0, 1, 0,. . . , 0),. . . , (0, 0, 0,. . . , 1)}

Dimension of Vector Space

The dimension of a vector space is the number of vectors in its basis, which represents the minimum number of independent directions needed to describe any vector in that space.

For example:

1. In a 2D plane, two basis vectors (like [1, 0] and [0, 1]) are needed, so the dimension is 2.
2. In 3D space, three basis vectors (like [1, 0, 0], [0, 1, 0], and [0, 0, 1]) are needed, so the dimension is 3.

The dimension tells us how many coordinates are required to specify any point in the space. It reflects the "size" or "complexity" of the space in terms of independent directions.

Application in Machine Learning: In machine learning vector spaces, basis, and dimension help us work with complex data. Basis vectors let us break down data into its most important directions, while dimension tells us how much information we need to represent that data. Techniques like dimensionality reduction (e.g., PCA) use these concepts to simplify data, reduce noise, and improve model performance by focusing on the key features.

Read More about Linear Algebra.