lbwei space

Introduction to Vectors

#Linear Algebra

Definition: A vector is a collection of numbers in \(\mathbb{R}\)

Operation

  • vector addition
    • element-wise addition
  • scalar multiplication

A linear combination consists of the two operations above

Axiom

  • Associativity of addition (結合)
  • Commutativity of addition (交換)
  • Distributivity of scalar multiplication (分配)
    1. with respect to vector addition
    2. with respect to field (scalar) addition
  • Compatibility of scalar multiplication (結合)
    • with field multiplication: \(a(b\boldsymbol{v}) = (ab)\boldsymbol{v}\)
  • Identity element of addition
    • exists an element \(\boldsymbol{0}\) (called zero vector) in V (vector space) such that \(\boldsymbol{v} + \boldsymbol{0} = \boldsymbol{v}\), for all \(\boldsymbol{v}\) in V
  • Identity element of scalar multiplication: \(1 \times \boldsymbol{v} = \boldsymbol{v}\)
    • \(1\) denotes the multiplicative identity in \(R\)
  • Inverse elements of addition
    • \(-\boldsymbol{v}\) is the additive inverse of \(\boldsymbol{v}\)

    另外有兩前提: (封閉性)

    1. \(x+y\in V\) whenever \(x, y \in V\)
    2. \(\alpha x\in V\) whenever \(\alpha \in F\)

見 Algebraic Extension。