Vector Space
• A vector space is a fundamental concept in mathematics and physics, particularly in linear algebra.
• It is a collection of objects called vectors, which can be added together and multiplied by scalars (real or complex numbers) and they remain objects of the same type.
• Vector spaces provide a framework for studying linear equations, transformations, and geometric concepts.
• All vectors of Vector space satisfy the following eight axioms.
Where ; u,v,w ∈ V (Vector space) and c,d ∈ F(Field)
In a vector space V over a field F:
• The elements of V are called vectors.
• The elements of F are called scalars.
Role of Field (F) in Vector Spaces
In a vector space V over a field F:
• The elements of V are called vectors.
• The elements of F are called scalars.
> If F=R, then V is a real vector space. (if F is a Real number)
> If F=C, then V is a complex vector space. (If F is a complex number)
Vector spaces are used in:
✔ Solving systems of linear equations.
✔ Studying linear transformations and matrices.
✔ Functional analysis (e.g., Hilbert spaces, Banach spaces).
✔ Machine learning (e.g., feature spaces, support vector machines).
✔ Physics (e.g., quantum mechanics, state spaces).
Question : Set of points satisfying X + Y = 0 forms a vector space ?
There are two points ( -2 , 2 ) and ( -4 , 4 ) on X + Y =0
Addition of two points : ( -2 , 2 ) + ( -4 , 4 ) = ( -6 , 6 )
( -6 , 6 ) is also on the line X + Y = 0
Multiply ( -2 , 2 ) by 2.5
2.5 * ( -2 , 2 ) = ( -5 , 5 )
( -5 , 5 ) also lies on X + Y =0
( 0 ,0 ) also lies on X + Y = 0
Inverse : Point A (-2 , 2 ) + ( 2 , -2 ) = ( 0 ,0 )
( 0 ,0 ) also lies on X + Y = 0
Identity : (0,0) + ( -2 , 2 ) = ( -2 , 2 )
Set of points satisfying X+Y= 0 forms a vector space:- Set follows Closure under Addition & Scalar Multiplication, Associativity, Commutativity, Existence of Additive Inverses, Distributivity of Scalar Multiplication, Associativity of Scalar Multiplication and Existence of a Multiplicative Identity.
Question : Set of points satisfying X + Y = 1 forms a vector space ?
The equation X + Y =1 defines a line in the XY-plane that does not pass through the origin. For a set to be a vector space, it must include the zero vector.
Alternate way :
if X = 0 and Y = 0 then:-
put X =0 & Y = 0 in equation X + Y = 1 :
0 + 0 ≠ 1
LHS(0) ≠ RHS (1)
Add point A & Point B
( 1 , 0 ) + ( 0 , 1 ) = ( 1 , 1 )
Point C ( 1 , 1 ) does not lie on X + Y = 1 , Therefore it is not a vector space
Addition is not closed: The sum of two solutions does not satisfy the equation.
Alternate :
points (0,1) and (1,0) are on X + Y = 1,
Sum of points (0,1) and (1,0) :-
(0,1) + (1,0) = (1,1)
let us check (1,1) lies on X + Y =1
Put X =1 & Y = 1 in equation:
LHS = 1 + 1 = 2
LHS ( value is 2 ) ≠ RHS (value is 1)
Scalar multiplication is not closed: Scaling a solution does not always satisfy the equation.
Point (1,0) multiply it by scalar c=6
6 * (1,0) = (6,0)
Put X =6 & Y = 0 in equation X + Y = 1:
LHS= 6 + 0 =6
LHS ( value is 6 ) ≠ RHS (value is 1)
Set of points satisfying X + Y = 1, does not form a Vector Space.
Examples
Vector Space (R1) : One Dimensional vector
A one-dimensional vector space R1 , consists of all real numbers, represented as scalars.
>> x Includes all real numbers ensuring closure under addition and scalar multiplication.
>> x Excludes +∞ and -∞ (since infinity is not a real number).
Example:
Vector Space (R2) : Two Dimensional vectors
Example: R2
Vector Space (R3) : Three Dimensional vectors
Real life Applications
Vector spaces provide a fundamental mathematical framework for solving real-world problems across disciplines. Their ability to represent and manipulate quantities with both magnitude and direction makes them invaluable in modern applications. Vector spaces have many real-life applications across various fields, from physics and engineering to finance and machine learning. Here are some practical examples:
Vector Space in Physics and Engineering
- Physics and Engineering
• Forces in Mechanics: When multiple forces act on an object, they are represented as vectors in a vector space. The resultant force is found using vector addition.
The resultant force (1N, 9N) moves the block in that direction. Since forces follow vector space properties (addition, scalar multiplication, etc.), they form a vector space in mechanics
Vector Space in Electromagnetic Field
Electric and Magnetic Fields: Electromagnetic fields are vector fields that define the force a charge experiences at any point in space.
• The force on the charged particle due to electric and magnetic fields is (26N, -12N, 0N).
• Since electromagnetic forces follow vector addition and scalar multiplication, they form a vector space in physics
Vector Space in Fluid Flow
• Fluid Flow: The velocity of a fluid at different points in space is represented using vector spaces.
Since velocity follows vector addition and scalar multiplication, it forms a vector space used in fluid dynamics simulations
Vector Space in Computer Graphics and Gaming
• 3D Modeling & Animation: Graphics software uses vector spaces to model object movements, transformations, and camera angles.
The movement doubles, showing scalar multiplication, a property of vector spaces. Since addition and scaling follow vector space properties, 3D modeling software like Blender, Maya, and Unreal Engine uses vector spaces to manipulate objects, animations, and camera angles
Vector Space in Ray Tracing in Rendering
• Ray Tracing in Rendering: Light rays are modeled as vectors, and their interactions with surfaces determine shading and reflections.
- The point (6, -1, 13) is where the ray reaches when travelling 4 units along its direction.
- This follows vector addition and scalar multiplication, satisfying vector space properties
Vector Space in Physics Engines
• Physics Engines: In games, motion, collision detection, and gravity simulations rely on vector space concepts.
Game engines use vector space properties (addition, subtraction, scaling) for movement, collision, and gravity in engines like Unreal, Unity, and Havok
Vector Space in Machine Learning and AI
Machine Learning and AI
• Feature Spaces: Data points in machine learning models exist in high-dimensional vector spaces. Each feature (height, weight, age, etc.) represents a dimension.
This shows how vector addition is valid in feature spaces.
Feature spaces form a vector space because they support addition, scalar multiplication, and distance measurement.
Machine learning models like KNN, PCA, and neural networks use vector spaces to find patterns in high-dimensional data
Vector Space in Word Embedding
• Word Embeddings in NLP: Words are represented as vectors in models like Word2Vec, allowing machines to understand semantic relationships.
This vector arithmetic shows how relationships between words exist in vector space.
- Word embeddings form a vector space since they support addition, subtraction, and similarity measurement.
- NLP models like Word2Vec, GloVe, and BERT use vector spaces to find relationships between words, enabling chatbots, translation, and sentiment analysis
Vector Space in Principal Component Analysis (PCA)
• Principal Component Analysis (PCA): PCA reduces the dimensions of large datasets while preserving important information.
PCA uses vector space operations (addition and scalar multiplication) to reduce dimensions for machine learning and analytics.
Vector Space in Economics and Finance
• Portfolio Optimization: Investments are treated as vectors, with their returns forming a vector space. Optimization algorithms maximize returns while minimizing risk.
Investments are represented as vectors in a financial vector space.
Portfolio optimization uses vector addition, scalar multiplication, and linear combinations to find the best investment mix.
Used in quantitative finance, risk management, and hedge fund strategies
Vector Space in Risk Analysis:
Financial risk factors are represented in a vector space to identify correlations and dependencies.
Risk factors are represented as vectors in a financial vector space.
Vector addition, scalar multiplication, and projections help measure risk exposure and correlation.
Used in banking, hedge funds, and investment risk management to minimize losses
Vector Space in Robotics and Autonomous Systems
• Motion Planning: Robots use vector spaces to calculate paths, avoid obstacles, and optimize movement.
• Control Systems: Feedback control in robotics relies on vector spaces for stability and precision.
Robots use vector spaces for motion planning (addition for path calculation) and control systems (error correction using subtraction and scaling).
Applied in autonomous vehicles, robotic arms, and drones for precise movement and obstacle avoidance
Vector Space in Navigation and GPS Systems
• Location Tracking: Latitude, longitude, and altitude define a position in a 3D vector space.
• Pathfinding Algorithms: Vector-based calculations determine optimal routes in GPS navigation.
Vector addition is used in GPS navigation to compute real-time positions.
Each movement step is calculated by adding velocity vectors to the current position.
This principle helps in Google Maps, Waze, and self-driving cars to optimize routes dynamically
Vector Space in Signal Processing and Communications
• Fourier Transforms: Signals are analyzed as vectors in frequency space for compression and filtering.
• Wireless Communication: Signal strength, phase, and direction are modeled using vector spaces.
Fourier Transforms and wireless signals use vector spaces for efficient analysis.
Applied in telecommunications, audio processing, and 5G networks
Vector Space in Medicine and Biology
• Medical Imaging (MRI, CT Scans): Image data is stored as vectors in multi-dimensional space for reconstruction.
• Genetics: DNA sequences are analyzed as high-dimensional vectors to detect genetic similarities.
Medical imaging and genetics use vector space operations (addition & scaling) for image processing and DNA analysis
Vector Space in Music and Audio Processing
• Sound Waves Representation: Audio signals are treated as vectors in time-frequency space.
• Auto-Tuning and Noise Reduction: Transformations in vector space help in improving audio quality.
Vector operations (addition, scaling, subtraction) enhance audio quality in music production and speech processing
Vector Space in Sports Analytics
• Player Performance Metrics: Athlete performance is represented in a multi-dimensional vector space for analysis and prediction.
• Trajectory Prediction: Ball or player movement in sports like soccer or basketball follows vector space principles.
Athlete performance metrics are vectors in multi-dimensional space for analysis & predictions.
Trajectory prediction uses vector addition for movement tracking.
Applied in sports analytics, AI coaching, and predictive models.
