Vector Spaces

vector space

Definition

A real vector space is a set \(V\) along with two binary operations \(+,\cdot\):

  • vector addition, \(+:V\times V\rightarrow V\)
  • scalar multiplication, \(\cdot: V\times \mathbb{R}\rightarrow V\)

such that the following axioms hold.

Addition axioms

  • Commutativity: \(u+v=v+u\) for all \(u,v\in V\)
  • Associativity: \(( u+v) +w=u+( v+w)\) for all \(u,v,w\in V\)
  • Additive identity: there exists an element in \(V\) denoted as \(0\) such that \(0+v=v\) for all \(v\in V\)
  • Additive inverse: for every \(v\in V\), there exists an element denoted as \(-v \in V\) such that \(v+( -v) =0\)

Scalar multiplication axioms

  • Associativity: \(( ab) v=a( bv)\) for all \(a,b\in \mathbb{R}\) and \(v\in V\)
  • Multiplicative identity: \(1v=v\) for all \(v\in V\)
  • Distributivity
    • \(( a+b) v=av+bv\) for all \(a,b\in \mathbb{R}\) and \(v\in V\)
    • \(a( u+v) =au+av\) for all \(u,v\in V\) and \(a\in \mathbb{R}\)

Examples

  • The set \(\mathbb{R}^{2}\) with the usual rules of addition and scalar multiplication is a vector space.
  • As an extension, the set \(\mathbb{R}^{n}\) with the usual rules of addition and scalar multiplication is a vector space.
  • The set of all polynomials of degree at most \(n\) with real coefficients is a vector space with the usual rules of polynomial addition and scalar multiplication. This is sometimes denoted as \(P_{n}(\mathbb{R})\).
  • The set of all real \(m\times n\) matrices is a vector space. This is sometimes denoted as \(M_{m\times n}(\mathbb{R})\).

Important points

  • Some of the familiar rules of algebra carry over:
    • \(av=0\Longrightarrow a=0\) or \(v=0\) or both, where \(a\in \mathbb{R}\) and \(v\in V\)
    • \(-v=( -1) v\), where \(v\in V\)
    • \(u+v=0\Longrightarrow u=-v\), where \(u,v\in V\)
  • For a given vector space, the zero element is unique. The zero element plays a very important role in the theory of vector spaces.
  • The same symbol is used to denote both the scalar zero and the vector zero. The context will make this distinction clear.
  • The zero element for some of the vector spaces discussed above:
    • For \(\mathbb{R}^{2}\), it is \(( 0,0)\).
    • For \(\mathbb{R}^{n}\), it is \(( 0,\cdots ,0)\)
    • For \(P_{2}(\mathbb{R})\) it is \(p( x) =0+0x+0x^{2}\), the zero polynomial.
    • For \(M_{2\times 2}(\mathbb{R})\), it is the zero matrix of size \(2\times 2\), \(\begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}\).