Generators and Generating set of a subspace in Linear Algebra: Definitions

Generators and Generating Set of a Subspace in Linear Algebra: Definitions

Generators and Generating Set of a Subspace in Linear Algebra

Generators and generating sets of a subspace are foundational concepts in Mathematics, specifically in Linear Algebra. By understanding these elements, a comprehensive grasp of how Vector Spaces are structured and expanded can be achieved. These topics are integral for analyzing vector spaces and exploring higher dimensions within Subspaces.

What You Will Learn?

Theorem-1: Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{\alpha\in V} \). Then the set \(\pmb{S=\{c\cdot \alpha: c\in F \} } \) is a subspace of \(\pmb{V} \).
Definition: Generator of a subspaces
Theorem-2: Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{\alpha,\beta\in V} \). Then the set \(\pmb{S=\{c\cdot\alpha+d\cdot\beta: c\in F \} } \) is a subspace of \(\pmb{V} \).
Definition: Generators of a subspaces
Theorem-3: Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{\alpha_{1},\alpha_{1},…,\alpha_{n}\in V} \). Then the set \(\pmb{S=\left\{\displaystyle\sum_{i=1}^{n}c_{i}\cdot\alpha_{i} : c_{i}\in F\right\} } \) is a subspace of \(\pmb{V} \).
Definition: Generating Set of a subspaces

Things to Remember

Before diving into this post, make sure you are familiar with: Basic Definitions and Concepts of

Mapping
Fields
Vector Space
Binary Composition
Subspace

Introduction

The study of generators and generating sets of a subspace emerged from efforts to formalize and organize the vast structures within vector spaces. Originally, this was developed to simplify vector combinations in physics and engineering. Today, generating sets offer a streamlined way to represent vector spaces efficiently, reducing the need for redundant elements. In addition, these concepts help mathematicians and scientists model complex systems, providing tools for data analysis, signal processing, and optimization. As the basis for understanding linear dependence and spanning, generators and generating sets are vital in fields ranging from machine learning to economics.

Theorem-1

Statement:

Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{\alpha\in V} \). Then the set \(\pmb{S=\{c\cdot \alpha: c\in F \} } \) is a subspace of \(\pmb{V} \).

  Proof:
  Given that \(\pmb{(V,+,\cdot)} \) is a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{\alpha\in V} \)
  To prove \(\pmb{S=\{c\cdot \alpha: c\in F \} } \) is a subspace of \(\pmb{V} \).

To prove \(\pmb{S\ne \Phi}\)
Since \(\pmb{0\in F} \)
\(\implies\pmb{0\cdot \alpha \in S} \)
\(\implies\pmb{\theta \in S} \)
\(\implies\pmb{S\ne \Phi} \)
To prove \(\pmb{\beta+\gamma\in S~\forall~\beta,\gamma\in S}\)
Let \(\pmb{\beta,\gamma\in S}\)
Then \(\pmb{\exists~ p,q\in F}\) such that \(\pmb{\beta=p\cdot \alpha,\gamma=q\cdot \alpha}\)
Now \(\pmb{\beta+\gamma=p\cdot \alpha+q\cdot \alpha}\)
\(\implies\pmb{\beta+\gamma=(p+q)\cdot \alpha}\)
\(\implies\pmb{\beta+\gamma\in S}\) since \(\pmb{p+q\in F} \)
To prove \(\pmb{d\cdot\delta\in S~\forall~\delta\in S, d\in F }\)
Let \(\pmb{\delta\in S, d\in F }\)
Then \(\pmb{\exists~ r\in F}\) such that \(\pmb{\delta=r\cdot \alpha}\)
Now \(\pmb{d\cdot\delta=d\cdot (r\cdot \alpha) }\)
\(\implies \pmb{d\cdot\delta=(d\cdot r)\cdot \alpha }\)
\(\implies \pmb{d\cdot\delta\in S }\) since \(\pmb{d\cdot r\in F} \)

Hence \(\pmb{S=\{c\cdot \alpha: c\in F \} } \) is a subspace of \(\pmb{V} \).

Generator of a subspace

Definition:

Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and and \(\pmb{\alpha\in V} \). Then the subspace \(\pmb{S=\{c\cdot \alpha: c\in F \} } \) of \(\pmb{V} \) is said to generated by the vector \(\pmb{\alpha} \) and the set is denoted by \(\pmb{L\{ \alpha\}}=\{c\cdot \alpha: c\in F \} \).

Theorem-2

Statement:

Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{\alpha_{1},\alpha_{2}\in V} \). Then the set \(\pmb{S=\{c_{1}\cdot\alpha_{1}+c_{2}\cdot\alpha_{2}: c_{i}\in F \} } \) is a subspace of \(\pmb{V} \).

  Proof:
  Given that \(\pmb{(V,+,\cdot)} \) is a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{\alpha_{1},\alpha_{2}\in V} \)
  To prove \(\pmb{S=\{c_{1}\cdot\alpha_{1}+c_{2}\cdot\alpha_{2}: c_{i}\in F \} } \) is a subspace of \(\pmb{V} \).

To prove \(\pmb{S\ne \Phi}\)
Since \(\pmb{0\in F} \)
\(\implies\pmb{0\cdot \alpha_{1}+0\cdot \alpha_{2} \in S} \)
\(\implies\pmb{\theta \in S} \)
\(\implies\pmb{S\ne \Phi} \)
To prove \(\pmb{\beta+\gamma\in S~\forall~\beta,\gamma\in S}\)
Let \(\pmb{\beta,\gamma\in S}\)
Then \(\pmb{\exists~ p_{1},p_{2},q_{1},q_{2}\in F}\) such that \(\pmb{\beta=p_{1}\cdot\alpha_{1}+p_{2}\cdot\alpha_{2}}\) and \(\pmb{\gamma=q_{1}\cdot\alpha_{1}+q_{2}\cdot\alpha_{2}}\)
Now \(\pmb{\beta+\gamma=(p_{1}\cdot\alpha_{1}+p_{2}\cdot\alpha_{2})+(q_{1}\cdot\alpha_{1}+q_{2}\cdot\alpha_{2})}\)
\(\implies\pmb{\beta+\gamma=(p_{1}+q_{1})\cdot \alpha_{1}+(p_{2}+q_{2})\cdot \alpha_{2}}\)
\(\implies\pmb{\beta+\gamma\in S}\) since \(\pmb{p_{1}+q_{1},p_{2}+q_{2}\in F} \)
To prove \(\pmb{d\cdot\delta\in S~\forall~\delta\in S, d\in F }\)
Let \(\pmb{\delta\in S, d\in F }\)
Then \(\pmb{\exists~ r_{1},r_{2}\in F}\) such that \(\pmb{\delta=r_{1}\cdot\alpha_{1}+r_{2}\cdot\alpha_{2}}\)
Now \(\pmb{d\cdot\delta=d\cdot (r_{1}\cdot\alpha_{1}+r_{2}\cdot\alpha_{2}) }\)
\(\implies \pmb{d\cdot\delta=(d\cdot r_{1})\cdot \alpha_{1} + (d\cdot r_{2})\cdot \alpha_{2} }\)
\(\implies \pmb{d\cdot\delta\in S }\) since \(\pmb{d\cdot r_{1},d\cdot r_{2}\in F} \)

Hence \(\pmb{S=\{c_{1}\cdot\alpha_{1}+c_{2}\cdot\alpha_{2}: c_{i}\in F \} } \) is a subspace of \(\pmb{V} \).

Generators of a subspace

Definition:

Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and and \(\pmb{\alpha_{1},\alpha_{2}\in V} \). Then the subspace \(\pmb{S=\{c_{1}\cdot\alpha_{1}+c_{2}\cdot\alpha_{2}: c_{i}\in F \} } \) of \(\pmb{V} \) is said to generated by the vectors \(\pmb{\alpha_{1},\alpha_{2}} \) and the set is denoted by \(\pmb{L\{ \alpha_{1},\alpha_{2}\}=\{c_{1}\cdot\alpha_{1}+c_{2}\cdot\alpha_{2}: c_{i}\in F \} }\).

Theorem-3

Statement:

Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{\alpha_{1},\alpha_{1},…,\alpha_{n}\in V} \). Then the set \(\pmb{S=\left\{\displaystyle\sum_{i=1}^{n}c_{i}\cdot\alpha_{i} : c_{i}\in F\right\} } \) is a subspace of \(\pmb{V} \).

  Proof:
  Given that \(\pmb{(V,+,\cdot)} \) is a vector space over a field \(\pmb{(F,+,\cdot)} \) and \(\pmb{\alpha_{1},\alpha_{1},…,\alpha_{n}\in V} \)
  To prove \(\pmb{S=\left\{\displaystyle\sum_{i=1}^{n}c_{i}\cdot\alpha_{i} : c_{i}\in F\right\} } \) is a subspace of \(\pmb{V} \).

To prove \(\pmb{S\ne \Phi}\)
Since \(\pmb{0\in F} \)
\(\implies\pmb{\displaystyle\sum_{i=1}^{n} 0\cdot\alpha_{i} \in S} \)
\(\implies\pmb{\displaystyle\sum_{i=1}^{n} \theta \in S} \)
\(\implies\pmb{\theta \in S} \)
\(\implies\pmb{S\ne \Phi} \)
To prove \(\pmb{\beta+\gamma\in S~\forall~\beta,\gamma\in S}\)
Let \(\pmb{\beta,\gamma\in S}\)
Then \(\pmb{\exists~ p_{i},q_{i}\in F,i=1,2,…,n}\) such that \(\pmb{\beta=\displaystyle\sum_{i=1}^{n}p_{i}\cdot\alpha_{i}}\) and \(\pmb{\gamma=\displaystyle\sum_{i=1}^{n}q_{i}\cdot\alpha_{i}}\)
Now \(\pmb{\beta+\gamma=\displaystyle\sum_{i=1}^{n}p_{i}\cdot\alpha_{i}+\displaystyle\sum_{i=1}^{n}q_{i}\cdot\alpha_{i}}\)
\(\implies\pmb{\beta+\gamma=\displaystyle\sum_{i=1}^{n}(p_{i}+q_{i})\cdot \alpha_{i}}\)
\(\implies\pmb{\beta+\gamma\in S}\) since \(\pmb{p_{i}+q_{i}\in F,i=1,2,…,n} \)
To prove \(\pmb{d\cdot\delta\in S~\forall~\delta\in S, d\in F }\)
Let \(\pmb{\delta\in S, d\in F,i=1,2,…,n }\)
Then \(\pmb{\exists~ r_{i}\in F}\) such that \(\pmb{\delta=\displaystyle\sum_{i=1}^{n}r_{i}\cdot\alpha_{i}}\)
Now \(\pmb{d\cdot\delta=d\displaystyle\sum_{i=1}^{n}r_{i}\cdot\alpha_{i} }\)
\(\implies \pmb{d\cdot\delta=\displaystyle\sum_{i=1}^{n}(d\cdot r_{i})\cdot\alpha_{i} }\)
\(\implies \pmb{d\cdot\delta\in S }\) since \(\pmb{d\cdot r_{i}\in F,i=1,2,…,n} \)

Hence \(\pmb{S=\left\{\displaystyle\sum_{i=1}^{n}c_{i}\cdot\alpha_{i} : c_{i}\in F\right\} } \) is a subspace of \(\pmb{V} \).

Generating Set of a subspace

Definition:

Let \(\pmb{(V,+,\cdot)} \) be a vector space over a field \(\pmb{(F,+,\cdot)} \) and and \(\pmb{\alpha_{1},\alpha_{1},…,\alpha_{n}\in V} \). Then the subspace \(\pmb{S=\left\{\displaystyle\sum_{i=1}^{n}c_{i}\cdot\alpha_{i} : c_{i}\in F\right\} } \) of \(\pmb{V} \) is said to generated by the vectors \(\pmb{\alpha_{1},\alpha_{2},…,,\alpha_{n}} \) and also the set \(\pmb{\alpha_{1},\alpha_{2},…,,\alpha_{n}} \) is said to be a Generating Set of \(\pmb{S}\)and it is denoted by \(\pmb{L\{ \alpha_{1},\alpha_{2},…,,\alpha_{n}\}=\left\{\displaystyle\sum_{i=1}^{n}c_{i}\cdot\alpha_{i} : c_{i}\in F\right\} }\).

Applications

Data Science and Machine Learning: Generating sets reduce high-dimensional data to its essential components, aiding in feature selection and dimensionality reduction.
Physics and Engineering: Generators are used to model subspaces in quantum mechanics and control theory, where specific states or ranges must be represented efficiently.
Computer Graphics: Vector spaces are managed using generating sets to optimize 3D rendering, improving computational efficiency.
Signal Processing: Generating sets enable efficient representation of signals and transformations, especially in Fourier transforms and wavelet analysis.

Conclusion

The Linear Sum of two subspaces is a powerful tool in Linear Algebra, extending beyond theory into numerous real-world applications. By understanding this concept, students gain a foundational skill essential for exploring advanced topics in Mathematics and its applications in fields like physics and computer science.

References

The concepts of generators and generating sets in Linear Algebra enable significant insights into the structure of vector spaces and subspaces. These ideas not only streamline mathematical calculations but also provide critical tools for applications in fields like data science, physics, and computer science. By understanding generating sets, one can simplify complex structures and model systems effectively, making them indispensable in advanced mathematics and practical applications alike.

Mappings
Binary Compositions
Vector Space
Linear Transformations

FAQs

What is a generator in Linear Algebra?
A generator is a vector used in linear combinations to span a subspace.
What defines a generating set of a subspace?
A generating set is a collection of vectors that span a subspace when combined linearly.
Why is a generating set important?
Generating sets simplify the representation of vector spaces and help in efficient calculations.
How is a generating set related to a basis?
A basis is a minimal generating set where the vectors are linearly independent.
What is the difference between a span and a generating set?
The span is the entire subspace formed by a generating set through linear combinations.
Can there be multiple generating sets for a subspace?
Yes, different sets of vectors can generate the same subspace.
How are generating sets applied in data science?
They help reduce data dimensions and identify essential features in high-dimensional spaces.
What role do generating sets play in physics?
They model state spaces in quantum mechanics, where specific states are represented.
Is every generating set also a basis?
No, only linearly independent generating sets are considered a basis.
How do generating sets simplify calculations in engineering?
They reduce complex systems to core elements, facilitating efficient modeling and analysis.