Subspaces

subspace

Definition

A subset \(\displaystyle U\) of a vector space \(\displaystyle V\) is called a subspace if it is a vector space with the same additive identity, addition and scalar multiplication as \(\displaystyle V\). To determine if a subset is a subspace, it is sufficient to check the following:

  • \(\displaystyle 0\in U\)

  • \(\displaystyle U\) is closed under vector addition

  • \(\displaystyle U\) is closed under scalar multiplication

Examples

  • \(\displaystyle \mathbb{R}^{2}\)
    • Trivial subspaces
      • \(\displaystyle \{( 0,0)\}\)
      • \(\displaystyle \mathbb{R}^{2}\)
    • Non-trivial subspaces
      • Any line passing through the origin
    • These are the only subspace of \(\displaystyle \mathbb{R}^{2}\)
  • \(\displaystyle \mathbb{R}^{3}\)
    • Trivial subspaces
      • \(\displaystyle \{( 0,0,0)\}\)
      • \(\displaystyle \mathbb{R}^{3}\)
    • Non-trivial subspaces
      • Any line passing through the origin
      • Any plane passing through the origin
    • These are the only subspaces of \(\displaystyle \mathbb{R}^{3}\)

Properties

  • If \(\displaystyle U\) and \(\displaystyle W\) are subspaces of \(\displaystyle V\), \(\displaystyle U\cap W\) is a subspace of \(\displaystyle V\)
  • If \(\displaystyle U\) and \(\displaystyle W\) are subspaces of \(\displaystyle V\), \(\displaystyle U\cup W\) needn’t be a subspace of \(\displaystyle V\)
    • Consider the x-axis and y-axis
    • The union is not closed under addition