Vector Spaces and Revision Session (Week 3 & Week 4)

Key Topics Covered

Definitions and Properties of Vector Spaces

• Vector Spaces: Set V with two operations: Vector addition and scalar multiplication.
• Closure properties:
  • Closed under vector addition: If V1, V2 ∈ V, then V1 + V2 ∈ V.
  • Closed under scalar multiplication: If α ∈ R and V ∈ V, then αV ∈ V.
• Conditions for Vector Spaces
  • Commutativity of addition
  • Associativity of addition
  • Existence of zero vector (identity element for addition)
  • Existence of additive inverse for each vector
  • Scalar multiplication properties including distributivity and associativity.

Verification of Vector Spaces

• Check closure properties first.
• Verify all eight axioms (related to addition and scalar multiplication).
• Examples provided:
  • Standard Euclidean space
  • Matrix spaces
  • Homogeneous system solutions

Subspaces

• Definition: Non-empty subset W of V that is also a vector space under same addition and scalar multiplication.
• Verification: Only check closure under addition and scalar multiplication; zero vector must be in W.
• Examples: R^2 and R^3 subspaces (lines and planes through origin).
• Special cases and properties:
  • Union of two subspaces need not be a subspace.
  • Intersection of two subspaces is always a subspace.
  • Sum of subspaces: U + W forms a subspace.

Exercises on Verification of Given Sets

• Analyze whether given sets with specific addition and scalar multiplication forms a vector space.
• Tests include commutativity, associativity, closure under operations, and specific vector space properties.

Linear Combinations, Spanning Sets, and Basis

• Linear Combination: Given vectors V1, V2, ..., Vn, any vector V = a1V1 + a2V2 + ... + anVn.
  • Example: Write (5,6) as a linear combination of (1,2) and (1,1).
• Span of a set: All possible linear combinations of a given set of vectors.
  • Example: Span of {(1,0),(0,1)} is R^2; Span of {(1,1,0), (0,1,-1)} is a plane in R^3.
• Basis: Linearly independent set of vectors that span the vector space.
  • Standard bases for R^n
  • Any n linearly independent vectors in R^n form a basis.

Dimension of Vector Spaces

• Defined as the cardinality of the basis set.
• Examples using simple vector spaces and matrix spaces.

Methods to Find Basis from Spanning Set

• Row Reduction Method: Write vectors as rows of a matrix and reduce to row echelon form; non-zero rows form the basis.
• Column Method: Write vectors as columns, perform row reduction, and identify pivot columns.

Rank of a Matrix

• Number of linearly independent rows or columns.
• Row rank and column rank are equal.
• Examples of calculating rank using row reduction, pivot columns, and simple observations for specific matrix forms.

Special Problems and Examples

• Exercises on determining rank, basis, and dimensions from given matrix and vector space conditions.

Additional Problem Solving

• Detailed steps for various problems including those with specific conditions on matrix entries and vector components.
• Verification of linear independence and spanning set qualifications.

Discussion and Q&A

• Handling specific student questions on topics such as proper subspaces, spanning set distinctions, and interpreting problem conditions.
• Emphasis on practical techniques for verifying conditions and solving related problems.

Study Tips

• Ensure understanding of the fundamental definitions and properties of vector spaces and subspaces.
• Practice problems on verifying vector spaces and subspaces, and finding linear combinations, spans, and bases.
• Review examples of rank calculations and basis reduction to solidify concepts.

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