Vector Spaces and Group Theory

Vector Spaces

• A non-empty set G with a binary operation * is a group if it satisfies four axioms:
  • Associative: (a * b) * c = a * (b * c)
  • Identity: There exists an identity element e such that a * e = e * a = a
  • Inverse: For each element a, there exists an element b such that a * b = e = b * a
• A group is abelian (or commutative) if a * b = b * a for all a, b in G.
• Examples of groups include:
  • The set of integers under addition (abelian group).
  • 2x2 matrices with real or complex entries under matrix addition (group).

Field Definition

• A field F is a non-empty set with two binary operations (+ and .) such that:

  • (F, +) is an abelian group.
  • (F \ {0}, .) is a multiplicative group.
  • Distributive property: a(b + c) = ab + ac and (a + b)c = ac + bc for all a, b, c in F.

• Examples of fields:

  • Real numbers with standard addition and multiplication.
  • Complex numbers with respect to addition and multiplication.

Vector Space Definition

• A vector space V over a field F satisfies:
  • (V, +) is an abelian group.
  • Closure under scalar multiplication: For every α in F and v in V, αv is in V.
  • Scalar multiplication properties:
    • α(v + w) = αv + αw
    • (α + β)v = αv + βv
    • α(βv) = (αβ)v
    • 1.v = v, where 1 is the identity in F.

Components and Types of Vectors

• An n-component vector a = (a1, a2, ..., an) is an ordered tuple of real numbers.
• Unit vector (ei): vector with unity in the ith component and zeros elsewhere.
• Null vector: a vector with all components zero.
• Sum vector: vector with unity as a value for each component, written as 1.
• Two vectors a and b are equal if their corresponding components are equal.
• The scalar product of vectors a and b is defined as the sum of the products of their components: ( a_1b_1 + a_2b_2 + ... + a_nb_n ).

Norm and Euclidean Space

• The norm of an n-component vector a = (a1, a2, ..., an) is given by ( ||a|| = \sqrt{a_1^2 + a_2^2 + ... + a_n^2} ).
• An n-dimensional Euclidean space En consists of all vectors of the form (a1, a2, ..., an).
• Vector addition and scalar multiplication in En are defined as:
  • Addition: (a1, a2, ..., an) + (b1, b2, ..., bn) = (a1 + b1, a2 + b2, ..., an + bn).
  • Scalar multiplication: λ(a1, a2, ..., an) = (λa1, λa2, ..., λan).

Subspaces and Linear Combinations

• A subset W of a vector space V is a subspace if W also forms a vector space over the same field.
• A linear combination of vectors {v1, v2, ..., vn} involves forming expressions of the form ( \alpha_1v_1 + \alpha_2v_2 + ... + \alpha_nv_n ).
• The linear span of a subset S of V is L(S), the set of all linear combinations of its elements.

Basis of a Vector Space

• A subset S is considered a basis of V if:
  • S consists of linearly independent elements.
  • S spans V (V = L(S)).
• Equivalent conditions for a basis include:
  • B is a minimal generating set of V.
  • B is a maximal set of linearly independent vectors.
  • Every vector in V can be uniquely expressed as a linear combination of vectors in B.

Examples of Basis

• Natural basis for R2 consists of vectors e1 = (1,0) and e2 = (0,1).
• The set {(1, 1), (−1, 2)} forms a basis for R2, verified through linear independence and spanning verification.

Properties and Theorems

• The maximum number of linearly independent vectors in En is n; any set of n+1 vectors is linearly dependent.
• A minimal spanning set is linearly independent and cannot be reduced without losing its spanning property.
• A maximal linearly independent set is a basis, as it spans the vector space while being linearly independent.

Independence Verification

• Vectors are linearly independent if the only solution to their linear combination equating to the zero vector is the trivial solution (all coefficients are zero).
• Verification involves solving a system of equations derived from vector combinations associated with a coefficient matrix.

Notable Results

• Any vector in Rn can be expressed as a linear combination of its basis vectors in only one unique way.
• All basis vectors in a finite-dimensional vector space share the same number of elements, linking their dimensionality directly to their independent properties.

This quiz covers the fundamental concepts of vector spaces, focusing on the definition of groups and their axioms. You'll explore the associative, identity, and inverse properties that define a binary operation on a set. Test your understanding of these essential algebraic structures.