Vector Space

Unit III: Vector Space

3.1 Introduction to Vector Spaces

A vector space is a mathematical structure consisting of a set of vectors with operations of addition and scalar multiplication, satisfying specific axioms. Vector spaces form the foundation of linear algebra and appear extensively in engineering, physics, computer science, and data analysis.

Think of familiar examples: the set of 2D vectors (x, y) in a plane, 3D vectors (x, y, z) in space, or the set of all polynomials of degree less than n. All of these are vector spaces.

3.2 Definition and Axioms

A vector space V over a field F (typically real numbers ℝ or complex numbers ℂ) is a set with two operations:

• Vector Addition: u + v is in V for any u, v ∈ V
• Scalar Multiplication: cv is in V for any c ∈ F and v ∈ V

Ten Vector Space Axioms:

3.3 Subspaces

A subspace W of a vector space V is a subset that is itself a vector space. To verify W is a subspace, check:

1. The zero vector 0 is in W
2. W is closed under addition
3. W is closed under scalar multiplication

3.4 Linear Combinations

A vector v is a linear combination of vectors v₁, v₂, ..., v_n if:

v = c₁v₁ + c₂v₂ + ... + c_nv_n

The span of vectors is the set of all possible linear combinations.

3.5 Linear Independence and Dependence

Vectors are linearly independent if the only solution to c₁v₁ + c₂v₂ + ... + c_nv_n = 0 is all c_i = 0.

3.6 Basis and Dimension

A basis for V is a set of linearly independent vectors that span V. The dimension equals the number of basis vectors.

• ℝ²: dimension = 2
• ℝ³: dimension = 3
• Polynomials of degree ≤ 2: dimension = 3

3.7 Coordinate Representation

With basis B = {b₁, ..., b_n}, every vector v has unique coordinates [c₁, ..., c_n] relative to B.