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
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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 : 1 v = v 1v = v 1 v = v
Zero scalar property : 0 v = 0 0v = 0 0 v = 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
∀ u , v ∈ W , ∀ c ∈ F , c u + v ∈ W \forall u, v \in W, \forall c \in \mathbb{F}, cu + v \in W ∀ u , v ∈ W , ∀ c ∈ F , c u + v ∈ 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 ∈ R 3 : x 1 + x 2 + x 3 > 0 {x \in \mathbb{R}^3 : x_1 + x_2 + x_3 > 0} x ∈ R 3 : x 1 + x 2 + x 3 > 0 not a subspace
Consider geometric interpretations
Subspaces as lines, planes, or hyperplanes through origin