Definition and Notation
Vector Concept
Vector: mathematical object with magnitude and direction. Represented as an ordered tuple of numbers. Dimension: number of components. Example: v = (v1, v2, ..., vn).
Notation
Common notations: boldface ( v ), arrow above ( \vec{v} ), or tuple notation (vi). Indexing: vi denotes the i-th component.
Geometric Interpretation
Vectors as directed line segments in Euclidean space. Origin to point position. Dimension corresponds to space dimension (2D, 3D, etc.).
Types of Vectors
Zero Vector
Zero vector (0): all components zero. No direction. Identity element for addition.
Unit Vector
Unit vector: magnitude equals 1. Direction indicator. Notation: \hat{v} = v / ||v||.
Position Vector
Position vector: from origin to a point. Defines point location in vector space.
Equal Vectors
Equal vectors: same magnitude and direction. Component-wise equality.
Vector Addition
Definition
Sum of two vectors: component-wise addition. If u = (u1, ..., un) and v = (v1, ..., vn), then u + v = (u1+v1, ..., un+vn).
Properties
Commutative: u + v = v + u. Associative: (u + v) + w = u + (v + w). Identity: u + 0 = u. Inverse: u + (-u) = 0.
Geometric Interpretation
Triangle rule: place tail of second vector at head of first; resultant vector from tail of first to head of second.
Scalar Multiplication
Definition
Multiplication of vector by scalar c ∈ ℝ: c·v = (c·v1, ..., c·vn).
Properties
Distributive: c(u + v) = cu + cv. Associative: (cd)v = c(dv). Identity: 1·v = v.
Effects on Magnitude and Direction
Magnitude scaled by |c|. Direction reversed if c < 0.
Vector Subtraction
Definition
Difference u - v defined as u + (-v). Component-wise: u - v = (u1 - v1, ..., un - vn).
Geometric Interpretation
Vector from tip of v to tip of u. Represents displacement between points.
Relation to Addition
Inverse operation of vector addition.
Dot Product (Scalar Product)
Definition
Dot product of vectors u and v: u · v = Σ uivi, sum of component-wise products.
Properties
Commutative: u · v = v · u. Distributive over addition: u · (v + w) = u · v + u · w. Scalar multiplication: (cu) · v = c(u · v).
Geometric Interpretation
u · v = ||u|| ||v|| cosθ, where θ is angle between vectors.
Applications
Projection, orthogonality (u · v = 0 means perpendicular), work calculation in physics.
u · v = u₁v₁ + u₂v₂ + ... + uₙvₙ||u|| = sqrt(u · u)cosθ = (u · v) / (||u|| ||v||)Cross Product (Vector Product)
Definition
Cross product defined only in ℝ³. For u = (u₁,u₂,u₃), v = (v₁,v₂,v₃), u × v = (u₂v₃ - u₃v₂, u₃v₁ - u₁v₃, u₁v₂ - u₂v₁).
Properties
Anticommutative: u × v = - (v × u). Distributive: u × (v + w) = u × v + u × w. Scalar multiplication: (cu) × v = c(u × v).
Geometric Interpretation
Result vector perpendicular to both u and v. Magnitude equals area of parallelogram spanned by u and v: ||u × v|| = ||u|| ||v|| sinθ.
Applications
Torque, angular momentum, normal vectors in geometry.
u × v =|i j k ||u₁ u₂ u₃||v₁ v₂ v₃|Magnitude (Length) of a Vector
Definition
Magnitude ||v|| = sqrt(v₁² + v₂² + ... + vₙ²).
Properties
Non-negative: ||v|| ≥ 0. Zero iff v is zero vector. Scalar multiplication effect: ||cv|| = |c| ||v||.
Unit Vector Calculation
Unit vector: \hat{v} = v / ||v||.
Vector Spaces and Subspaces
Vector Space Definition
Set V with vector addition and scalar multiplication satisfying 8 axioms: associativity, commutativity, identity, inverse, distributivity, compatibility, and scalar identity.
Examples
ℝⁿ, polynomial spaces, function spaces.
Subspaces
Subset W ⊆ V closed under addition and scalar multiplication. Contains zero vector.
Span
Span of set S = all linear combinations of vectors in S. Span(S) subspace of V.
Linear Independence
Definition
Set of vectors {v₁,...,vₖ} linearly independent if c₁v₁ + ... + cₖvₖ = 0 implies all cᵢ = 0.
Dependence
Vectors dependent if non-trivial linear combination equals zero.
Testing Methods
Row reduction on matrix formed by vectors. Zero rows indicate dependence.
Significance
Linear independence essential for basis, dimension determination.
Basis and Dimension
Basis Definition
Set of linearly independent vectors that span vector space V.
Dimension
Number of vectors in any basis of V. Unique for finite-dimensional spaces.
Examples
Standard basis in ℝⁿ: e₁=(1,0,...,0),..., eₙ=(0,0,...,1).
Change of Basis
Representing vectors in different bases via coordinate transformations.
| Vector Space | Dimension | Basis Example |
|---|---|---|
| ℝ² | 2 | {(1,0), (0,1)} |
| ℝ³ | 3 | {(1,0,0), (0,1,0), (0,0,1)} |
| P₂ (polynomials degree ≤ 2) | 3 | {1, x, x²} |
Applications of Vectors
Physics
Force, velocity, acceleration represented as vectors. Work: dot product of force and displacement.
Computer Graphics
Modeling positions, directions, normals. Transformations with vectors and matrices.
Engineering
Statics, dynamics, signal processing use vector representations.
Data Science
Vectors represent data points in multidimensional space. Used in machine learning algorithms.
Mathematics
Solving linear systems, eigenvalue problems, optimization.
References
- Axler, S., Linear Algebra Done Right, Springer, 3rd ed., 2015, pp. 1-250.
- Strang, G., Introduction to Linear Algebra, Wellesley-Cambridge Press, 5th ed., 2016, pp. 45-180.
- Lay, D. C., Linear Algebra and Its Applications, Pearson, 5th ed., 2015, pp. 30-210.
- Anton, H., Rorres, C., Elementary Linear Algebra, Wiley, 11th ed., 2013, pp. 10-200.
- Friedberg, S. H., Insel, A. J., Spence, L. E., Linear Algebra, Prentice Hall, 4th ed., 2003, pp. 25-220.