Vector spaces are the backbone of linear algebra, providing a framework for understanding linear relationships. They're defined by specific rules called axioms, which govern how vectors interact through addition and multiplication.

Subspaces are subsets of vector spaces that follow the same rules. They're crucial for breaking down complex problems into simpler parts. Understanding subspaces helps you grasp the structure of vector spaces and solve linear equations more effectively.

Vector Spaces and Properties

Definition and Axioms

• Vector space comprises a set V of vectors with two operations
  • Vector addition
  • Scalar multiplication
• Satisfies specific axioms over a field (typically real numbers ℝ or complex numbers ℂ)
• Ten vector space axioms ensure mathematical structure
  • Closure under addition and scalar multiplication
  • Commutativity and associativity of addition
  • Existence of zero vector
  • Existence of additive inverses
  • Distributivity of scalar multiplication over vector addition
  • Distributivity of scalar multiplication over field addition
  • Scalar identity property: $1v = v$
  • Zero scalar property: $0v = 0$

Dimensions and Examples

• Dimension denotes number of vectors in the basis
  • Basis consists of linearly independent vectors spanning the entire space
• Common vector space examples
  • ℝⁿ (n-dimensional real space)
  • Function spaces (continuous functions on an interval)
  • Polynomial spaces (polynomials of degree ≤ n)
• Abstract vector spaces extend beyond geometric vectors
  • Vectors can represent functions, matrices, or other mathematical objects

Verifying Vector Spaces

Axiom Verification Process

• Check all ten vector space axioms for given set and operations
• Test closure property for addition and scalar multiplication
  • Ensure operations result in elements within the set
• Verify commutativity and associativity of vector addition
  • Use arbitrary elements from the set
• Demonstrate existence of unique zero vector and additive inverses
• Confirm distributive properties
  • Scalar multiplication over vector addition
  • Scalar multiplication over field addition
• Pay special attention to scalar identity and zero scalar properties

Disproving Vector Spaces

• Use counter-examples to disprove vector space status
• Identify specific axiom violations
  • Example: Set of positive real numbers fails zero vector axiom
• Analyze edge cases and boundary conditions
  • Example: Set of integers under real scalar multiplication not closed

Subspaces of Vector Spaces

Subspace Definition and Properties

• Subspace consists of subset W of vector space V
  • Forms vector space under same operations as V
• Three conditions for subspace verification
  • W is non-empty
  • W is closed under vector addition
  • W is closed under scalar multiplication
• Zero vector of original space must be in subspace
  • Serves as additive identity for subspace
• Common subspace types
  • Null spaces (solutions to Ax = 0)
  • Column spaces (span of matrix columns)
  • Row spaces (span of matrix rows)

Subspace Relationships

• Intersection of two subspaces always forms subspace
• Union of subspaces generally not a subspace
  • Exception: One subspace contained within the other
• Span of vector set from original space always creates subspace
• Subspace test combines three conditions into single statement
  • $\forall u, v \in W, \forall c \in \mathbb{F}, cu + v \in W$
  • F represents the underlying field (ℝ or ℂ)

Subspace Identification

Verification Process

• Check three subspace conditions
  • Non-emptiness
  • Closure under addition
  • Closure under scalar multiplication
• Verify zero vector presence in subset
  • Proves non-emptiness
  • Partially addresses closure
• Test addition closure
  • Show sum of arbitrary elements remains in subset
• Demonstrate scalar multiplication closure
  • Prove scalar multiple of any element stays in subset

Special Cases and Considerations

• Analyze sets defined by equations or conditions
  • Show conditions preserved under vector space operations
• Use counter-examples to disprove subspace status efficiently
  • Example: Plane not passing through origin fails zero vector condition
• Examine subsets defined by strict inequalities
  • Often fail subspace criteria due to zero vector exclusion
  • Example: ${x \in \mathbb{R}^3 : x_1 + x_2 + x_3 > 0}$ not a subspace
• Consider geometric interpretations
  • Subspaces as lines, planes, or hyperplanes through origin