Vector Spaces and Axioms

Vector Space

• A vector space is defined by two operations: vector addition and scalar multiplication.
• For any elements x, y in vector space V, x + y is also an element of V (closure under addition).
• For any scalar a from field F and vector x in V, the product ax is an element of V (closure under scalar multiplication).

Axioms of Vector Spaces

• Eight axioms must be satisfied for a set to be a vector space:
  • Commutativity of addition: x + y = y + x
  • Associativity of addition: (x + y) + z = x + (y + z)
  • Existence of zero vector: x + 0 = x
  • Existence of additive inverses: x + (-x) = 0
  • Multiplicative identity: 1 * x = x
  • Associativity of multiplication: (ab)x = a(bx)
  • Distributive property: a(x + y) = ax + ay
  • Scalar addition: (a + b)x = ax + bx

Subspaces

• A subset W of a vector space V is a subspace if:
  • W is also a vector space over F.
  • Operations of addition and scalar multiplication in W match those in V.
• The zero subspace is {0}, a subspace in any vector space.

Linear Combinations and Span

• A vector is a linear combination of a subset S if it can be expressed using scalars from F.
• The span of a subset S, denoted span(S), includes all possible linear combinations of vectors in S.
• The span of the empty set is {0}.

Linear Dependence and Independence

• A subset is linearly dependent if there exist scalars, not all zero, such that a linear combination equals zero.
• A subset is linearly independent if the only solution to their linear combination equaling zero is the trivial solution (all coefficients are zero).
• If a set contains the zero vector, it is linearly dependent.

Basis and Dimension

• A basis is a linearly independent subset that spans the vector space.
• The standard basis for F^n consists of unit vectors, like e1 = (1,0,0,...), e2 = (0,1,0,...).
• The dimension of a vector space is the number of vectors in any basis, denoted dim(V).
• Finite-dimensional spaces have a basis consisting of a finite number of vectors; infinite-dimensional spaces do not.

The Replacement Theorem

• If V is spanned by n vectors, and L is a linearly independent set with m vectors, then m ≤ n. Additionally, it is possible to find n-m vectors from the set G such that their union with L spans V.

Transforming a Set into a Basis

• To convert a finite set S into a basis:
  • Show it generates V.
  • Select a nonzero vector from S.
  • Exclude scalar multiples of the chosen vector from the basis.
  • Add new vectors iteratively, ensuring linear independence.

Dimension Relations

• The dimension of any subspace W cannot exceed that of vector space V, expressed as dim(W) ≤ dim(V).
• If dim(W) = dim(V), then W and V are the same space.

Matrices Context

• The dimension of a matrix space Mmxn(F) is mn.
• Each matrix entry is determined by its row (m) and column (n) positions.