Subspaces

Definition

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

• \(0\in U\)

• \(U\) is closed under vector addition

• \(U\) is closed under scalar multiplication

Examples

• \(\mathbb{R}^{2}\)
  • Trivial subspaces
    • \(\{( 0,0)\}\)
    • \(\mathbb{R}^{2}\)
  • Non-trivial subspaces
    • Any line passing through the origin
  • These are the only subspace of \(\mathbb{R}^{2}\)
• \(\mathbb{R}^{3}\)
  • Trivial subspaces
    • \(\{( 0,0,0)\}\)
    • \(\mathbb{R}^{3}\)
  • Non-trivial subspaces
    • Any line passing through the origin
    • Any plane passing through the origin
  • These are the only subspaces of \(\mathbb{R}^{3}\)
• \(\mathcal{M}_{3 \times 3}(\mathbb{R})\)
  • Trivial subspaces
    • \(\{0\}\), the set containing the zero matrix
    • \(\mathcal{M}_{3 \times 3}(\mathbb{R})\)
  • Some non-trivial subspaces
    • The set of symmetric matrices
    • The set of triangular matrices

Subspace sum

The sum of two subspaces \(U, W\) of a vector space \(V\) is denoted as \(U + W\) and is defined as:

\[ U + W := \{u + w\ :\ u \in U, w \in W\} \]

\(U + W\) is the set of all vectors that are of the form \(u + w\), where \(u\) comes from \(U\) and \(w\) from \(V\). The subspace sum is the generalization of the idea of union of two sets to two vector spaces. \(U + W\) is a subspace of \(V\).

Properties

• If \(U\) and \(W\) are subspaces of \(V\), \(U\cap W\) is a subspace of \(V\)
• If \(U\) and \(W\) are subspaces of \(V\), \(U \cap W\) is always non-empty. It should at least contain the zero vector.
• If \(U\) and \(W\) are subspaces of \(V\), \(U\cup W\) needn’t be a subspace of \(V\)
  • Consider the x-axis and y-axis
  • The union is not closed under addition
• \(U + W\) is the smallest subspace of \(V\) that contains both \(U\) and \(W\).