1.2 Subspaces and their properties

Vector spaces are the foundation of linear algebra. Subspaces are special subsets that inherit the structure of the parent space. They're crucial for understanding how vector spaces are organized and how they relate to each other.

Subspaces must be closed under addition and scalar multiplication, and contain the zero vector. This section dives into their properties, examples, and how to identify them. It also compares the behavior of subspace intersections and unions.

Subspaces of Vector Spaces

Definition and Key Properties

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• Subspace W of vector space V constitutes a subset of V that forms a vector space under inherited vector addition and scalar multiplication operations
• Three essential properties define a subspace:
  • Closure under addition ensures u + v belongs to W for any vectors u and v in W
  • Closure under scalar multiplication guarantees cv remains in W for any vector v in W and scalar c
  • Zero vector of V must be an element of W
• Subspaces inherit vector space axioms from parent space (commutativity, associativity, distributivity)
• Improper subspaces include trivial subspace {0} and entire vector space V
• Dimension of subspace always less than or equal to dimension of parent vector space

Examples and Non-Examples

• Common subspace examples
  • Spans of vectors (set of all linear combinations of given vectors)
  • Null spaces of linear transformations
  • Solution sets of homogeneous systems of linear equations
• Non-subspace examples
  • Affine subspaces (translations of vector subspaces)
  • Open sets in vector spaces
  • Subsets lacking zero vector
• Infinite-dimensional vector spaces may require advanced techniques for subspace verification (general elements, proof by contradiction)

Identifying Subspaces

Verification Process

• Prove subset W as subspace by confirming three defining properties
  • Closure under addition
  • Closure under scalar multiplication
  • Containment of zero vector
• Systematic approach involves arbitrary elements
  • Consider u and v in W, prove u + v remains in W
  • Take v in W and scalar c, show cv stays in W
  • Demonstrate zero vector belongs to W
• Counterexamples disprove subspace status
  • Find specific vectors or scalars violating subspace properties
  • Example: Show sum of two elements leaves the subset

Practical Applications

• Span verification as subspace
  • Prove linear combinations of spanning vectors remain in span
• Null space of linear transformation as subspace
  • Show vectors mapped to zero form a subspace
• Solution set of homogeneous linear equations as subspace
  • Demonstrate solutions satisfy subspace properties
• Polynomial subspaces
  • Verify set of polynomials with specific degree or property forms subspace

Intersection vs Union of Subspaces

Intersection Properties

• Intersection of subspaces always forms a subspace
• Proof outline for intersection as subspace:
  • Take vector in intersection, belongs to all constituent subspaces
  • Show closure under addition using properties of individual subspaces
  • Demonstrate closure under scalar multiplication
  • Zero vector, present in all subspaces, exists in intersection
• Applications of intersection property
  • Finding common solutions to multiple linear systems
  • Identifying shared characteristics of different vector subspaces

Union Behavior

• Union of subspaces generally not a subspace
• Exception occurs when one subspace contains the other
• Counterexample construction
  • Use two distinct one-dimensional subspaces (lines through origin in $\mathbb{R}^2$)
  • Show sum of vectors from different subspaces may not belong to either
• Implications of union behavior
  • Limits direct combination of subspaces
  • Necessitates alternative methods for combining vector subspace properties

Comparative Analysis

• Intersection preserves subspace structure, union often does not
• Intersection finds common elements, union aggregates all elements
• Practical consequences
  • Intersection useful for identifying shared properties
  • Union requires caution in vector space analysis
• Importance in vector space decomposition
  • Direct sum concept relates to union behavior
  • Intersection crucial in determining independence of subspaces