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Introduction to Vector Space

The Vector Space

We are now moving from concrete lists of numbers to the abstract definition of a vector space defined over R\mathbb{R} (Real numbers).

The 10 Fundamental Axioms

For a set VV to be a Vector Space, it must satisfy 10 axioms under two operations: Vector Addition and Scalar Multiplication.

  1. Closure (Addition & Multiplication)
  2. Associativity
  3. Commutativity
  4. Identity (Existence of Zero Vector 0\mathbf{0} )
  5. Inverse (Additive Inverse)
  6. …and distributive properties.
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Subspaces

A Subspace is a non-empty subset WW of a vector space VV that is itself a vector space under the same operations.

The Subspace Test:

  1. Is 0∈W\mathbf{0} \in W?
  2. Is WW closed under addition?
  3. Is WW closed under scalar multiplication?

Key Properties

  • The zero vector is unique.
  • The additive inverse of any vector is unique.